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Auf dem Ten-sor-pro-dukt $\OML/cmm/m/it/12 X \OMS/c msy/m/n/12 [] \OML/cmm/m/it/12 Y$ \OT1/cmr/m/n/12 ist dann ka-no-nisch ei-ne $\OML/cmm/m/i t/12 S$\OT1/cmr/m/n/12 -Linksmodulstruktur [] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1809. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1809. [29] Overfull \hbox (66.23329pt too wide) detected at line 1901 [] [] Overfull \hbox (35.13168pt too wide) detected at line 1909 \OML/cmm/m/it/12 r \OMS/cmsy/m/n/12 ^^A [] \OT1/cmr/m/n/12 = [] []\OML /cmm/m/it/12 r \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 k[] [] \OMS/cmsy/m/n/12 2 \O ML/cmm/m/it/12 X[] \OMS/cmsy/m/n/12 ^^B \OML/cmm/m/it/12 X[] \OMS/cmsy/m/n/12 ^ ^B \OML/cmm/m/it/12 ::: \OMS/cmsy/m/n/12 ^^B \OML/cmm/m/it/12 X[] [] Overfull \hbox (66.23329pt too wide) detected at line 1941 [] [] [30] [31] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 2081. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\new@ifnextchar' on input line 2081. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 2081. Overfull \hbox (3.89703pt too wide) in paragraph at lines 2085--2090 []\OT1/cmr/m/n/12 Wir ha-ben oben f[]ur einen Ring $\OML/cmm/m/it/12 R;$ \OT1/c mr/m/n/12 einen $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Rechtsmodul $\OML/cmm/m/it /12 X[]$ \OT1/cmr/m/n/12 und einen $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Linksmo dul [] [32] Overfull \hbox (1.89522pt too wide) in paragraph at lines 2111--2115 \OT1/cmr/m/n/12 Auf der abel-schen Grup-pe $[][] []$ ist dann ka-no-nisch ei-ne $\OML/cmm/m/it/12 T$\OT1/cmr/m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (6.90138pt too wide) in paragraph at lines 2161--2182 \OT1/cmr/m/n/12 al-ler $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Linksmodulhomomorph ismen aus ei-nem $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Linksmodul $\OML/cmm/m/it /12 X$ \OT1/cmr/m/n/12 in einen $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Linksmodul [] [33]pdfTeX warning (ext4): destination with the same identifier (name{equation. 1.1}) has been already used, duplicate ignored \relax l.2219 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2219 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2219 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2219 \begin{equation} pdfTeX warning (ext4): destination wi th the same identifier (name{equation.1.1}) has been already used, duplicate ig nored \relax l.2229 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2229 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2229 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2229 \begin{equation} pdfTeX warning (ext4): destination wi th the same identifier (name{equation.1.1}) has been already used, duplicate ig nored \relax l.2238 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2238 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2238 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2238 \begin{equation} [34] Overfull \hbox (13.80779pt too wide) detected at line 2288 [] [] Underfull \hbox (badness 1881) in paragraph at lines 2333--2333 \OT1/cmr/m/n/10 aus $\OML/cmm/m/it/10 ^^Q [] \OT1/cmr/m/n/10 = 1[]$ folgt), und die Kom-mu-ta-ti-vit[]at des Dia-gramms (1.3) ist []aquivalent zu [] [35]pdfTeX warning (ext4): destination with the same identifier (name{equation. 1.1}) has been already used, duplicate ignored \relax l.2377 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2377 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2377 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.2377 \begin{equation} [36] [37] Overfull \hbox (18.98582pt too wide) detected at line 2547 [] [] [38] [39] [40] [41] Overfull \hbox (8.50015pt too wide) in paragraph at lines 2760--2762 []\OT1/cmr/bx/n/12 Beispiele: \OT1/cmr/m/n/12 Wir wol-len kurz zwei be-kann-te Spe-zi-alf[]alle von Mo-noi-dal-ge-bren erw[]ahnen, [] Overfull \hbox (36.92903pt too wide) detected at line 2784 [] \OT1/cmr/m/n/12 ; [] Overfull \hbox (4.61238pt too wide) in paragraph at lines 2784--2788 \OT1/cmr/m/n/12 die Mul-ti-pli-ka-ti-on auf die-sem Mo-no-id wer-de durch Hin-t er-ein-an-der-schrei-ben von W[]ortern [] Overfull \hbox (32.32428pt too wide) in paragraph at lines 2834--2838 []\OT1/cmr/m/n/12 Die ka-no-ni-sche Ab-bil-dung $[] : \OML/cmm/m/it/12 A \OMS/c msy/m/n/12 ! \OML/cmm/m/it/12 A\U/msb/m/n/12 ^^^\OML/cmm/m/it/12 I;$ $a \OMS/cm sy/m/n/12 7! []$ \OT1/cmr/m/n/12 ist dann ein $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/ 12 -Algebrahomomorphismus [] [42] Overfull \hbox (2.84962pt too wide) detected at line 2875 \OML/cmm/m/it/12 k [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 k [] \U/msb/m/n/12 ^^^ [] [] [43] [44] [45] [46] Overfull \hbox (6.8832pt too wide) detected at line 3245 [] [] [47] Overfull \hbox (32.44115pt too wide) detected at line 3291 [] [] Overfull \hbox (33.09392pt too wide) detected at line 3309 [] [] Overfull \hbox (45.2187pt too wide) detected at line 3350 [] [] [48]pdfTeX warning (ext4): destination with the same identifier (name{equation. 1.1}) has been already used, duplicate ignored \relax l.3433 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3433 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3433 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3433 \begin{equation} [49] [50] Overfull \hbox (2.35999pt too wide) in paragraph at lines 3601--3609 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 ein kom-mu-ta-ti-ver Ring und $\OML/cmm/m/it/12 X$ \OT1/cmr/m/n/12 ein $\OML/cm m/m/it/12 k$\OT1/cmr/m/n/12 -Modul. Dann k[]onnen wir ei-ne nat[]urliche [] [51]pdfTeX warning (ext4): destination with the same identifier (name{equation. 1.1}) has been already used, duplicate ignored \relax l.3701 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3701 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3701 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3701 \begin{equation} [52]pdfTeX warning (ext4): destinati on with the same identifier (name{equation.1.1}) has been already used, duplica te ignored \relax l.3706 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3706 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3706 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.3706 \begin{equation} [53] Overfull \hbox (12.3914pt too wide) detected at line 3828 [] [] [54] [55] Overfull \hbox (1.32796pt too wide) in paragraph at lines 3988--3992 []\OT1/cmr/m/n/12 Wir k[]onnen die Re-sul-ta-te von 1.19. und 1.19$[]$. ver-all -ge-mei-nern, in-dem wir zus[]atzlich [] [56] [57] [58] Overfull \hbox (6.36638pt too wide) in paragraph at lines 4296--4305 [][]$ \OT1/cmr/m/n/12 durch $\OML/cmm/m/it/12 ' \OT1/cmr/m/n/12 = [][] \OML/cmm /m/it/12 ;$ \OT1/cmr/m/n/12 wo-bei f[]ur je-des $\OML/cmm/m/it/12 M \OMS/cmsy/m /n/12 2 []$ \OT1/cmr/m/n/12 der $\OML/cmm/m/it/12 S$\OT1/cmr/m/n/12 -Linksmodul isomorphismus [] [59] [60] Overfull \hbox (11.22623pt too wide) in paragraph at lines 4382--4390 \OT1/cmr/m/n/12 In Ka-pi-tel 1 ha-ben wir den Be-griff ei-ner $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Algebra auf drei ver-schie-de-ne (aber []aquivalente) [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.4399 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4399 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4399 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4399 \begin{equation} pdfTeX warning (ext4): destination wi th the same identifier (name{equation.1.1}) has been already used, duplicate ig nored \relax l.4409 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4409 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4409 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4409 \begin{equation} pdfTeX warning (ext4): destination wi th the same identifier (name{equation.1.1}) has been already used, duplicate ig nored \relax l.4416 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4416 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4416 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.4416 \begin{equation} Overfull \hbox (5.82077pt too wide) in paragraph at lines 4433--4437 []\OT1/cmr/m/it/12 Bemerkung: \OT1/cmr/m/n/12 F[]ur je-de Coal-ge-bra $[]$ ist die Ab-bil-dung $^^A$ in-jek-tiv (denn $[] \OMS/cmsy/m/n/12 ^^N [] [61] [62] [63] Overfull \hbox (64.52069pt too wide) detected at line 4649 [] [] Overfull \hbox (29.26227pt too wide) detected at line 4669 [] [] [64] [65] [66] Overfull \hbox (26.39369pt too wide) in paragraph at lines 4818--4824 \OT1/cmr/m/n/12 des Ten-sor-pro-duk-tes meh-re-rer Vek-torr[]aume) ei-ne li-nea -re Ab-bil-dung $^^A[] : [] \OMS/cmsy/m/n/12 ! [] Overfull \hbox (3.66818pt too wide) detected at line 4899 [] [] [67] Overfull \hbox (3.66818pt too wide) detected at line 4929 [] [] Overfull \hbox (19.84763pt too wide) detected at line 4969 [] [] [68] Overfull \hbox (44.81671pt too wide) detected at line 5002 [] [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 : [] Overfull \hbox (48.56078pt too wide) detected at line 5029 [] [] [69] Overfull \hbox (19.51826pt too wide) detected at line 5076 [] [] Overfull \hbox (26.39369pt too wide) in paragraph at lines 5076--5082 \OT1/cmr/m/n/12 des Ten-sor-pro-duk-tes meh-re-rer Vek-torr[]aume) ei-ne li-nea -re Ab-bil-dung $^^A[] : [] \OMS/cmsy/m/n/12 ! [] [70] Overfull \hbox (35.32362pt too wide) detected at line 5109 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.5160 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5160 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5160 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5160 \begin{equation} [71] Overfull \hbox (22.14255pt too wide) detected at line 5311 [] [] [72] Overfull \hbox (9.87526pt too wide) in paragraph at lines 5360--5363 []\OT1/cmr/m/n/12 Wir k[]onnen (2.10) fol-gen-der-ma-^^Yen um-schrei-ben: Ist $ \OML/cmm/m/it/12 T$ \OT1/cmr/m/n/12 ei-ne end-li-che Men-ge nat[]urlicher [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.5363 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5363 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5363 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5363 \begin{equation} Overfull \hbox (62.86728pt too wide) detected at line 5477 [] [] [73] [74] Overfull \hbox (22.93832pt too wide) detected at line 5508 [] [] [75] Overfull \hbox (9.42592pt too wide) in paragraph at lines 5518--5531 \OT1/cmr/m/n/12 Somit ha-ben wir $[] [] = [] []$ [] Overfull \hbox (33.78467pt too wide) detected at line 5591 [] [] [76] [77]pdfTeX warning (ext4): destination with the same identifier (name{equa tion.1.1}) has been already used, duplicate ignored \relax l.5641 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5641 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5641 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5641 \begin{equation} Overfull \hbox (20.6521pt too wide) in paragraph at lines 5650--5658 []\OT1/cmr/bx/n/12 4) \OT1/cmr/m/n/12 Sei-en $\OML/cmm/m/it/12 n \OMS/cmsy/m/n/ 12 ^^U \OT1/cmr/m/n/12 0$ und $\OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 2 []$ \OT1/c mr/m/n/12 be-lie-big. Dann ist $^^A[] = [] \OMS/cmsy/m/n/12 ^^N [] [78] Overfull \hbox (36.03891pt too wide) detected at line 5705 [] [] [79 ] [80 ] Overfull \hbox (2.36288pt too wide) in paragraph at lines 5806--5810 []\OT1/cmr/m/n/12 Nach Be-mer-kung 2.1$[]$ \OT1/cmr/bx/n/12 5) \OT1/cmr/m/n/12 (an-ge-wandt auf $\OML/cmm/m/it/12 u \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ` \OMS/ cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 1$ und $\OML/cmm/m/it/12 v \OT1/cmr/m/n/12 = \O ML/cmm/m/it/12 n \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 `$\OT1/cmr/m/n/12 ) ist $[] \OMS/cmsy/m/n/12 ^^N [] Overfull \hbox (4.0851pt too wide) in paragraph at lines 5817--5824 []\OT1/cmr/m/n/12 Nach Be-mer-kung 2.1$[]$ \OT1/cmr/bx/n/12 5) \OT1/cmr/m/n/12 (an-ge-wandt auf $\OML/cmm/m/it/12 u \OT1/cmr/m/n/12 = 0$ und $\OML/cmm/m/it/12 v \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 2$) ist $[] \OMS/cmsy/m/n/12 ^^N [] Overfull \hbox (4.0851pt too wide) in paragraph at lines 5825--5831 []\OT1/cmr/m/n/12 Nach Be-mer-kung 2.1$[]$ \OT1/cmr/bx/n/12 5) \OT1/cmr/m/n/12 (an-ge-wandt auf $\OML/cmm/m/it/12 u \OT1/cmr/m/n/12 = 1$ und $\OML/cmm/m/it/12 v \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 2$) ist $[] \OMS/cmsy/m/n/12 ^^N [] [81]pdfTeX warning (ext4): destination with the same identifier (name{equation. 1.1}) has been already used, duplicate ignored \relax l.5870 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5870 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5870 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.5870 \begin{equation} Overfull \hbox (14.37085pt too wide) in paragraph at lines 5896--5901 []\OT1/cmr/m/it/12 Induktionsanfang (der In-duk-ti-on nach $\OML/cmm/m/it/12 `$ \OT1/cmr/m/it/12 ): \OT1/cmr/m/n/12 F[]ur $\OML/cmm/m/it/12 ` \OT1/cmr/m/n/12 = 0$ ist $^^A[] = [] \OMS/cmsy/m/n/12 ^^N [] [82] Overfull \hbox (23.36516pt too wide) detected at line 5992 \OT1/cmr/m/n/12 ^^A[] = [] \OMS/cmsy/m/n/12 ^^N \OT1/cmr/m/n/12 ^^A[] = [] \OMS /cmsy/m/n/12 ^^N \OT1/cmr/m/n/12 ^^A[]\OML/cmm/m/it/12 : [] Overfull \hbox (9.9491pt too wide) detected at line 6079 [] [] [83] [84] [85] Overfull \hbox (24.53123pt too wide) detected at line 6271 [] [] Overfull \hbox (13.69644pt too wide) detected at line 6289 [] [] [86] [87] [88] Overfull \hbox (14.17662pt too wide) detected at line 6510 \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 ::: \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 ^^A [] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 ::: \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 ::: \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x[] []x \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 C: [] Overfull \hbox (7.13469pt too wide) detected at line 6531 [] [] Overfull \hbox (1.22763pt too wide) in paragraph at lines 6531--6548 \OT1/cmr/m/n/12 Diese For-mel be-sagt im we-sent-li-chen, da^^Y wir ein in der sum-men-lo-sen Sweedler-Notation [] [89] Overfull \hbox (31.55046pt too wide) detected at line 6562 [] [] Overfull \hbox (12.84901pt too wide) in paragraph at lines 6613--6626 \OT1/cmr/m/n/12 dung $\OML/cmm/m/it/12 f \OT1/cmr/m/n/12 : [] \OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 X;$ \OT1/cmr/m/n/12 da^^Y der Term $\OML/cmm/m/it/12 f []$ \ OT1/cmr/m/n/12 als $[] \OML/cmm/m/it/12 f []$ [] [90] Overfull \hbox (0.53467pt too wide) in paragraph at lines 6638--6639 \OT1/cmr/m/n/12 Folgende ein-fa-che Be-mer-kung ist ein Bei-spiel f[]ur die An- wen-dung der Sweedler-Notation: [] [91] Overfull \hbox (36.46475pt too wide) detected at line 6741 [] [] [92] Overfull \hbox (26.99364pt too wide) detected at line 6810 [] [] Overfull \hbox (20.32707pt too wide) detected at line 6832 [] [] [93] Overfull \hbox (27.5581pt too wide) detected at line 6860 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.6879 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6879 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6879 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6879 \begin{equation} pdfTeX warning (ext4): destination wi th the same identifier (name{equation.1.1}) has been already used, duplicate ig nored \relax l.6890 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6890 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6890 \begin{equation} pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.6890 \begin{equation} [94] Overfull \hbox (13.33855pt too wide) detected at line 6942 [] [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 6972. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\new@ifnextchar' on input line 6972. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\left' on input line 6972. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\right' on input line 6972. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 6972. [95] Overfull \hbox (1.41998pt too wide) detected at line 7038 [] [] Overfull \hbox (4.95169pt too wide) detected at line 7069 [] [] [96] [97] Overfull \hbox (26.99573pt too wide) detected at line 7216 [] [] Overfull \hbox (26.28706pt too wide) detected at line 7226 [] [] [98] [99] [100] Overfull \hbox (12.18253pt too wide) detected at line 7463 \OT1/cmr/m/n/12 ^^A [] = \OML/cmm/m/it/12 c[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 d[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 c[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 d[]\OT1/cmr/m/n/12 ; \OML/cmm/m/it/12 " [] \OT1/cmr /m/n/12 = \OML/cmm/m/it/12 " [] " [] []c \OMS/cmsy/m/n/12 2 \OML/cmm/ m/it/12 C[]d \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 D: [] [101] [102] Overfull \hbox (8.37686pt too wide) detected at line 7632 [] [] \OML/cmm/m/it/10 g[] \OMS/cmsy/m/n/10 \OML/cmm/m/it/10 g[] \OT1/cmr/m/n/10 = [] \OMS/cmsy/m/n/10 ^^@ [] \OT1/cmr/m/n /10 = [] \OML/cmm/m/it/10 ^^L[]g[] \OMS/cmsy/m/n/10 \OML/cmm/m/it/10 g[] \OMS/cmsy/m/n/10 ^^@ [] \OML/cmm/m/it/10 ^^L[]g[] \OMS/cm sy/m/n/10 \OML/cmm/m/it/10 g[] \OT1/cmr/m/n/10 = 0\OML/cmm/m/it/10 : [] [103] Overfull \hbox (18.04984pt too wide) detected at line 7696 [] [] [104] Overfull \hbox (38.13496pt too wide) in paragraph at lines 7748--7762 \OT1/cmr/m/n/12 also $[] [] = 0$ und da-mit $[] [] = [] []$ [] Overfull \hbox (29.92812pt too wide) in paragraph at lines 7748--7762 \OT1/cmr/m/n/12 (denn $\OML/cmm/m/it/12 ^^Y \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 ^^Y$ \OT1/cmr/m/n/12 ist li-ne-ar). Doch we-gen $[] [] = [] \ OMS/cmsy/m/n/12 []$ \OT1/cmr/m/n/12 und $[] [] = [] [105] Overfull \hbox (20.81612pt too wide) detected at line 7907 [] [] [106] Overfull \hbox (0.1283pt too wide) in paragraph at lines 7907--7912 \OT1/cmr/m/n/12 also $[][] [] = \OML/cmm/m/it/12 G [] :$ \OT1/cmr/m/n/12 Da $[] $ bi-jek-tiv ist, ist al-so $[] [] = [] [] \OML/cmm/m/it/12 ;$ [] [107] Overfull \hbox (44.46402pt too wide) detected at line 7977 []\OML/cmm/m/it/12 ; [] Overfull \hbox (12.00272pt too wide) detected at line 8036 [] [] [108] [109] Overfull \hbox (2.75409pt too wide) in paragraph at lines 8102--8109 []\OT1/cmr/m/n/12 Wir wer-den im Fol-gen-den ei-ne Rei-he von Ei-gen-schaf-ten von Hopfal-ge-bren er-gr[]unden; [] Overfull \hbox (3.69064pt too wide) in paragraph at lines 8152--8159 \OML/cmm/m/it/12 G []$\OT1/cmr/m/n/12 ) ist ei-ne Un-ter-al-ge-bra von $\OML/cm m/m/it/12 H$ \OT1/cmr/m/n/12 (ge-nau-er ge-sagt: man kann die Hopfal-ge-bra $\O ML/cmm/m/it/12 k []$ [] [110] [111] [112] [113] Overfull \hbox (25.7554pt too wide) detected at line 8478 [] [] [114] [115] Overfull \hbox (9.49586pt too wide) detected at line 8686 [] [] Overfull \hbox (32.32973pt too wide) in paragraph at lines 8717--8722 []\OT1/cmr/bx/n/12 2) a) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/ 12 ei-ne Bi-al-ge-bra, und $\OML/cmm/m/it/12 C$ \OT1/cmr/m/n/12 ei-ne co-kom-mu -ta-ti-ve Coal-ge-bra. Dann ist $[] [] = [] Overfull \hbox (31.87267pt too wide) in paragraph at lines 8723--8728 []\OT1/cmr/bx/n/12 b) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne Hopfal-ge-bra, und $\OML/cmm/m/it/12 C$ \OT1/cmr/m/n/12 ei-ne co-kom-mu-t a-ti-ve Coal-ge-bra. Dann ist $[] [] = [] [116] Overfull \hbox (4.66653pt too wide) detected at line 8755 [] [] Overfull \hbox (0.14027pt too wide) in paragraph at lines 8765--8768 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/m/n/12 Erst-mal ist $\OML/cmm/m/it/12 fS$ \ OT1/cmr/m/n/12 ein Al-ge-bra-ho-mo-mor-phis-mus, denn $[] [] = \OML/cmm/m/it/12 f [] \OT1/cmr/m/n/12 = [] [117] Overfull \hbox (9.62633pt too wide) detected at line 8870 [] [] Overfull \hbox (8.16682pt too wide) detected at line 8933 [] [] [118] Overfull \hbox (3.87383pt too wide) detected at line 8943 [] [] [119] Overfull \hbox (7.43004pt too wide) detected at line 9017 [] [] \OT1/cmr/m/n/12 = [] [] [] [] [] = \OML/cmm/m/it/12 x \OT1/cmr/m/n/12 = [] [] \OML/cmm/m/it/12 : [] Overfull \hbox (32.00203pt too wide) detected at line 9128 [] [] [120] Overfull \hbox (46.14658pt too wide) detected at line 9159 [] [] [121] [122] [123] [124] [125] Overfull \hbox (39.2241pt too wide) in paragraph at lines 9576--9583 []\OT1/cmr/m/n/12 Da $\OML/cmm/m/it/12 "[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 "[]$ \OT1/cmr/m/n/12 und $[]$ Al-ge-bra-ho-mo-mor-phis-men si nd, ist auch die Ver-ket-tung $[] \OMS/cmsy/m/n/12 ^^N []$ [] Overfull \hbox (17.74933pt too wide) detected at line 9649 [] [] [126] [127] [128] [129] [130] Overfull \hbox (7.15776pt too wide) in paragraph at lines 9980--9983 []\OT1/cmr/bx/n/12 6) \OT1/cmr/m/n/12 Ei-ne \OT1/cmr/m/it/12 Vor-be-mer-kung \O T1/cmr/m/n/12 zu den n[]achsten Bei-spie-len: In der Bi-al-ge-bra $\OML/cmm/m/i t/12 H \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 k [][]$ [] [131] Overfull \hbox (12.2273pt too wide) detected at line 10117 [] [] [132] Overfull \hbox (59.75925pt too wide) detected at line 10152 [] [] [133] Overfull \hbox (17.51093pt too wide) detected at line 10215 [] [] [134] Overfull \hbox (3.55057pt too wide) detected at line 10271 [] [] Overfull \hbox (13.1198pt too wide) in paragraph at lines 10300--10305 \OT1/cmr/m/n/12 Dieses $[]$ fak-to-ri-siert []uber $\OML/cmm/m/it/12 H$ \OT1/cm r/m/n/12 (wo-bei wir $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 als Fak-toral-ge-bra der frei-en Al-ge-bra $\OML/cmm/m/it/12 k []$ [] [135] [136] Overfull \hbox (16.17014pt too wide) detected at line 10423 [] [] [137] Overfull \hbox (11.97614pt too wide) in paragraph at lines 10513--10519 \OT1/cmr/m/n/12 auf die-se Po-ly-no-me) de-fi-nie-ren. (Hier ver-wen-den wir, d a^^Y $[][] \OML/cmm/m/it/12 ;$ $[][]\OT1/cmr/m/n/12 !$ und $[][]$ tats[]achlich [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.10530 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.10530 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.10530 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.10530 \begin{equation} Overfull \hbox (49.54828pt too wide) detected at line 10574 [] [] [138] Overfull \hbox (12.40071pt too wide) detected at line 10601 [][] \U/msa/m/n/12 ^^] [] \OML/cmm/m/it/12 ; [] [139] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 10627. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\new@ifnextchar' on input line 10627. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 10627. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 10627. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\new@ifnextchar' on input line 10627. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 10627. [140] [141] [142] [143] [144] [145] [146] [147] Overfull \hbox (43.92738pt too wide) detected at line 11242 [] [] Overfull \hbox (60.19408pt too wide) detected at line 11266 [] [] [148] Overfull \hbox (6.06906pt too wide) detected at line 11332 [] [] [149] [150] Overfull \hbox (42.96278pt too wide) detected at line 11399 [] [] [151] Overfull \hbox (1.95132pt too wide) in paragraph at lines 11450--11457 []\OT1/cmr/m/n/12 Wir k[]onnen die-se De-fi-ni-ti-on noch ein we-nig um-for-mu- lie-ren, in-dem wir die $\U/msb/m/n/12 Z$\OT1/cmr/m/n/12 -bilineare [] [152] Overfull \hbox (3.23967pt too wide) in paragraph at lines 11540--11545 \OT1/cmr/m/n/12 (Dabei ist wie-der $[]$ ist ein Iso-mor-phis-mus. Die $\OML/cmm /m/it/12 k$\OT1/cmr/m/n/12 -Algebrahomomorphismen [] Overfull \hbox (4.25252pt too wide) in paragraph at lines 11570--11577 \OML/cmm/m/it/12 v \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^V []$ \OT1/cmr/m/n/12 f[]ur al-le $\OML/cmm/m/it/12 \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 k$ \OT1/cmr /m/n/12 und $\OML/cmm/m/it/12 v \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 V$ \OT1/cmr /m/n/12 de-fi-niert. Die-se $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Vektorraumstru ktur [] [153] Overfull \hbox (5.48181pt too wide) in paragraph at lines 11671--11674 []\OT1/cmr/m/n/12 Genug der Tri-via-lit[]aten, jetzt wer-den wir (wie ver-spro- chen) $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulstrukturen [] [154] [155] Overfull \hbox (6.29723pt too wide) in paragraph at lines 11747--11749 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Ist $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne Bi-al-ge-bra, dann ist die oben de-fi-nier-te $\OML/cmm/m/it/12 "$\OT1/cm r/m/n/12 -Modulstruktur auf $\OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 tats[]achlich [] [156] [157] Overfull \hbox (12.05727pt too wide) in paragraph at lines 11954--11962 []\OT1/cmr/bx/n/12 a) \OT1/cmr/m/n/12 Ge-nau dann ist f[]ur al-le $\OML/cmm/m/i t/12 U; V; W \OMS/cmsy/m/n/12 2 []$ \OT1/cmr/m/n/12 der ka-no-ni-sche $\OML/cmm /m/it/12 k$\OT1/cmr/m/n/12 -Vektorraumisomorphismus [] Overfull \hbox (2.60547pt too wide) in paragraph at lines 11988--11995 \OT1/cmr/m/it/12 auch wenn $\OML/cmm/m/it/12 H$ \OT1/cmr/m/it/12 ei-ne Hopfal-g e-bra ist\OT1/cmr/m/n/12 ! Auch der ka-no-ni-sche $\OML/cmm/m/it/12 k$\OT1/cmr/ m/n/12 -Vektorraumhomomorphismus [] [158] [159] [160] [161] Overfull \hbox (9.97986pt too wide) in paragraph at lines 12348--12352 []\OT1/cmr/bx/n/12 4) \OT1/cmr/m/n/12 Auch die in 2.18. \OT1/cmr/bx/n/12 3) \OT 1/cmr/m/n/12 de-fi-nier-te Hopfal-ge-bra $\OML/cmm/m/it/12 k [] \U/msb/m/n/12 ^ ^^ []$ \OT1/cmr/m/n/12 ist ei-ne zu-sam-menh[]angende [] [162] [163] [164] [165]pdfTeX warning (ext4): destination with the same identif ier (name{equation.1.1}) has been already used, duplicate ignored \relax l.12608 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.12608 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.12608 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.12608 \begin{equation} Overfull \hbox (3.89856pt too wide) detected at line 12666 [] [] Overfull \hbox (4.77226pt too wide) in paragraph at lines 12666--12675 \OT1/cmr/m/n/12 also $\OML/cmm/m/it/12 ^^Y[] [] \OT1/cmr/m/n/12 = 0$. Das hei^^ Yt, $\OML/cmm/m/it/12 ^^Y[] [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^Y[] []$\OT1 /cmr/m/n/12 . We-gen $\OML/cmm/m/it/12 ^^Y[] [] \OT1/cmr/m/n/12 = [] [166] Overfull \hbox (17.14883pt too wide) detected at line 12794 [] [] [167] [168] Overfull \hbox (1.32332pt too wide) detected at line 12863 [] [] [169] Overfull \hbox (10.48143pt too wide) detected at line 12980 [] [] [170] Overfull \hbox (20.05554pt too wide) detected at line 13021 [] [] \OT1/cmr/m/n/12 = [] [] = ^^H[] [] = [] + \OML/cmm/m/it/12 v \OMS/cmsy/m/ n/12 ^^A [] [] \OML/cmm/m/it/12 : [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.13049 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13049 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13049 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13049 \begin{equation} [171] Overfull \hbox (11.42773pt too wide) detected at line 13087 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.13095 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13095 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13095 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13095 \begin{equation} Overfull \hbox (13.99036pt too wide) in paragraph at lines 13108--13113 []\OT1/cmr/m/n/12 F[]ur je-des $\OML/cmm/m/it/12 i \OMS/cmsy/m/n/12 2 []$ \OT1/ cmr/m/n/12 ist $\OML/cmm/m/it/12 x[]$ \OT1/cmr/m/n/12 pri-mi-tiv (nach \OT1/cmr /bx/n/12 2)\OT1/cmr/m/n/12 , da $\OML/cmm/m/it/12 x[] \OMS/cmsy/m/n/12 2 \OML/c mm/m/it/12 H[]$\OT1/cmr/m/n/12 ), al-so $[]$-schiefprimitiv. [] Overfull \hbox (0.49577pt too wide) detected at line 13166 [] [] [172] Overfull \hbox (31.36353pt too wide) in paragraph at lines 13170--13174 []\OT1/cmr/m/n/12 Anwendung von (2.51) auf $\OML/cmm/m/it/12 ` \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 1$ lie-fert $[] [] = []$ [] [173] Overfull \hbox (8.501pt too wide) in paragraph at lines 13260--13272 \OT1/cmr/m/n/12 bra. Wir wer-den wei-ter un-ten in Bei-spiel 2.60. \OT1/cmr/bx/ n/12 1) \OT1/cmr/m/n/12 ei-ne wei-te-re Hopfal-ge-bra $[]$ [] [174] Overfull \hbox (3.55328pt too wide) in paragraph at lines 13315--13318 []\OT1/cmr/m/n/12 Beispielsweise ist al-so $[][] = \OML/cmm/m/it/12 v[]v[]:::v[ ]$ \OT1/cmr/m/n/12 f[]ur je-des $\OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 2 \U/msb/m /n/12 N$ \OT1/cmr/m/n/12 und f[]ur al-le $\OML/cmm/m/it/12 v[]; v[]; :::; v[] \ OMS/cmsy/m/n/12 2 [] [175]pdfTeX warning (ext4): destination with the same identifier (name{equation .1.1}) has been already used, duplicate ignored \relax l.13423 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13423 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13423 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13423 \begin{equation} Overfull \hbox (0.30609pt too wide) in paragraph at lines 13434--13437 []\OT1/cmr/m/it/12 Induktionsanfang: \OT1/cmr/m/n/12 F[]ur $\OML/cmm/m/it/12 ^^ W \OT1/cmr/m/n/12 = 0$ ist $[] [] \OML/cmm/m/it/12 ::: [] \OT1/cmr/m/n/12 = [] = [] Overfull \hbox (21.98491pt too wide) in paragraph at lines 13451--13469 \OT1/cmr/m/n/12 enth[]alt, oder ei-ne Teil-men-ge von $[]$, wel-che $\OML/cmm/m /it/12 n$ \OT1/cmr/m/n/12 nicht enth[]alt) und $\OMS/cmsy/m/n/12 P[] [] \ [] Overfull \hbox (11.18262pt too wide) in paragraph at lines 13481--13485 []\OT1/cmr/m/n/12 Nun ist $\OMS/cmsy/m/n/12 P[] [] \OT1/cmr/m/n/12 = \OMS/cmsy/ m/n/12 P []$ \OT1/cmr/m/n/12 (denn die Teil-men-gen von $[]$, [] [176] Overfull \hbox (43.31819pt too wide) detected at line 13569 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.13573 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13573 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13573 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13573 \begin{equation} [177] [178]pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13746 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13746 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13746 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13746 \begin{equation} [179]pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13787 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13787 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13787 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13787 \begin{equation} [180] Overfull \hbox (22.7078pt too wide) in paragraph at lines 13860--13864 []\OT1/cmr/m/n/12 Anwendung von (2.70) auf $\OML/cmm/m/it/12 n \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 $ \OT1/cmr/m/n/12 und $[] = []$ er-gibt $\OML/cmm/m/it/12 ^ ^P [] \OT1/cmr/m/n/12 = [] Overfull \hbox (18.38379pt too wide) in paragraph at lines 13877--13881 []\OT1/cmr/m/n/12 Wegen $\OML/cmm/m/it/12 a \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 z[]z[]:::z[]$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 b \OT1/cmr/m/n/12 = \OML/cm m/m/it/12 z[]z[]:::z[]$ \OT1/cmr/m/n/12 ist nun $\OML/cmm/m/it/12 ab \OT1/cmr/m /n/12 = [] [] = [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.13898 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13898 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13898 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.13898 \begin{equation} [181] Overfull \hbox (37.75539pt too wide) detected at line 13952 [] [] [182] [183] [184] [185] Overfull \hbox (5.86467pt too wide) in paragraph at lines 14274--14277 []\OT1/cmr/m/n/12 Bekanntlich ist $[] = []$ ei-ne gra-du-ier-te $\OML/cmm/m/it/ 12 k$\OT1/cmr/m/n/12 -Algebra. [] Overfull \hbox (1.04733pt too wide) in paragraph at lines 14284--14291 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/m/n/12 Wir m[]ussen zei-gen, da^^Y $[]$ ei- ne gra-du-ier-te $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Coalgebra [] pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1} ) has been already used, duplicate ignored \relax l.14306 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14306 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14306 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14306 \begin{equation} Overfull \hbox (27.60266pt too wide) detected at line 14358 [] [] [186]pdfTeX warning (ext4): destination with the same identifier (name{equation .1.1}) has been already used, duplicate ignored \relax l.14376 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14376 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14376 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.14376 \begin{equation} Overfull \hbox (10.35231pt too wide) in paragraph at lines 14418--14426 []\OT1/cmr/bx/n/12 l) \OT1/cmr/m/n/12 Be-trach-ten wir die $\OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 -Bialgebra $\OML/cmm/m/it/12 T[]V \OT1/cmr/m/n/12 = []$. Laut \ OT1/cmr/bx/n/12 j) \OT1/cmr/m/n/12 ist $[]$ [] Overfull \hbox (0.55403pt too wide) in paragraph at lines 14418--14426 \OT1/cmr/m/n/12 ei-ne gra-du-ier-te $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Algebr a, und laut \OT1/cmr/bx/n/12 k) \OT1/cmr/m/n/12 ist $[]$ ei-ne gra-du-ier-te $\ OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Coalgebra. [] Overfull \hbox (48.60718pt too wide) in paragraph at lines 14427--14432 []\OT1/cmr/m/n/12 Laut Satz 2.45 \OT1/cmr/bx/n/12 6) \OT1/cmr/m/n/12 (an-ge-wan dt auf $[] = []$) ist al-so $[]$ [] Overfull \hbox (12.14798pt too wide) in paragraph at lines 14433--14440 \OT1/cmr/m/n/12 ten wir di-rekt die Aus-sa-ge von Bei-spiel 2.50 \OT1/cmr/bx/n/ 12 4) \OT1/cmr/m/n/12 (denn f[]ur $[] = []$ [] [187] [188] [189] [190] [191] Overfull \hbox (36.15341pt too wide) detected at line 14776 [][] [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 a[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 a[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 ::: \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 a[] []a[]; a[]; :::; a[] \OMS/cmsy/m/n/12 2 \OML/cmm /m/it/12 V: [] Overfull \hbox (13.9919pt too wide) in paragraph at lines 14776--14785 \OT1/cmr/m/n/12 (Diese Ab-bil-dung $[][]$ ist hier-durch wohl-de-fi-niert, denn die rech-te Sei-te $[] \OML/cmm/m/it/12 a[] \OMS/cmsy/m/n/12 [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 14803. 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[192] Overfull \hbox (6.44173pt too wide) in paragraph at lines 14866--14869 []\OT1/cmr/bx/n/12 1) \OT1/cmr/m/n/12 Sind $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 A[]$ \OT1/cmr/m/n/12 zwei Al-ge-bren, und sind $\OML/cmm /m/it/12 ^^[; ^^\ \OMS/cmsy/m/n/12 2 [] [] \OML/cmm/m/it/12 ;$ \OT1/cmr/m/n/12 und ist $\OML/cmm/m/it/12 ^^N \OMS/cmsy/m/n/12 2 [] [] \OML/cmm/m/it/12 ;$ [] Overfull \hbox (0.50471pt too wide) in paragraph at lines 14891--14896 []\OT1/cmr/bx/n/12 c) \OT1/cmr/m/n/12 Ge-nau dann gilt $\OML/cmm/m/it/12 '[] [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 '[] [] '[] []$ \OT1/cmr/m/n/12 f[]ur al-le $\OML/cmm/m/it/12 a; b \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 , wenn $\OML/cmm/m/it/12 ^^N [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^[ [] ^^N [] \OT1/cmr/m/n/12 + [] Overfull \hbox (19.41127pt too wide) detected at line 14917 \OML/cmm/m/it/12 '[] [] '[] [] \OT1/cmr/m/n/12 = [] [] = [] [] [193] Overfull \hbox (20.2667pt too wide) detected at line 15008 \OML/cmm/m/it/12 H[] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 : [] [194] [195] [196] [197] [198] Overfull \hbox (23.99594pt too wide) in paragraph at lines 15466--15472 \OT1/cmr/m/n/12 macht, wird durch $\OML/cmm/m/it/12 g \OMS/cmsy/m/n/12 7! []$ \ OT1/cmr/m/n/12 ein-deu-tig fest-ge-legt. Da^^Y $\OML/cmm/m/it/12 k []$ \OT1/cmr /m/n/12 ei-ne $\OML/cmm/m/it/12 k []$\OT1/cmr/m/n/12 -Linksmodulalgebra [] Overfull \hbox (1.65363pt too wide) in paragraph at lines 15473--15476 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 q \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 von $0$ ver-schie-den, und sei $\OML/cmm/ m/it/12 H \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 k [] :$ [] [199] LaTeX Font Info: Trying to load font information for U+euf on input line 155 45. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 2013/01/14 v3.01 Euler Fraktur ) [200] [201] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 15740. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 15740. Overfull \hbox (35.97107pt too wide) in paragraph at lines 15794--15797 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 Wir be-zeich-nen die Ka-te-go-rie der $\O ML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomoduln (mit $\OML/cmm/m/it/12 C$\OT1/ cmr/m/n/12 -Comodulhomomorphismen [] [202] [203] Overfull \hbox (24.93356pt too wide) in paragraph at lines 15898--15903 \OT1/cmr/m/n/12 (denn $[] [] = \OML/cmm/m/it/12 W[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 C;$ \OT1/cmr/m/n/12 weil $[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 C$ \OT1/cmr/m/n/12 ka-no-nisch iso-morph zu $[] \U/msb/m/n/12 ^^^ []$ [] Overfull \hbox (2.21777pt too wide) detected at line 15953 [] [] [204] Overfull \hbox (3.37213pt too wide) in paragraph at lines 15971--15976 []\OT1/cmr/m/n/12 Die Axio-me ei-nes Co-mo-duls m[]ussen nur auf ei-ner Ba-sis des Vek-tor-raums $\OML/cmm/m/it/12 V$ []\OT1/cmr/m/n/12 uberpr[]uft [] [205] Overfull \hbox (25.45221pt too wide) in paragraph at lines 16061--16064 \OT1/cmr/m/n/12 definiert ist. Fer-ner be-trach-ten wir auf dem Vek-tor-raum $\ OML/cmm/m/it/12 H \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 die $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -R echtscomodulstruktur [] Overfull \hbox (2.90697pt too wide) in paragraph at lines 16074--16080 \OT1/cmr/m/n/12 che gleich-zei-tig $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmo dulendomorphismen und $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulendomor phismen [] Overfull \hbox (14.72821pt too wide) in paragraph at lines 16107--16112 []\OT1/cmr/m/n/12 uberhaupt wohl-de-fi-niert ist, d. h. da^^Y $^^H[] \OMS/cmsy/ m/n/12 2 []$ \OT1/cmr/m/n/12 f[]ur je-des $\OML/cmm/m/it/12 f \OMS/cmsy/m/n/12 2 [][]$ [] [206] [207] Overfull \hbox (20.4317pt too wide) detected at line 16322 [] [] [208] Overfull \hbox (7.44312pt too wide) in paragraph at lines 16341--16346 \OT1/cmr/m/n/12 ist wohl-de-fi-niert (gem[]a^^Y \OT1/cmr/bx/n/12 (a)\OT1/cmr/m/ n/12 ), ein Al-ge-bra-ho-mo-mor-phis-mus (gem[]a^^Y \OT1/cmr/bx/n/12 (b)\OT1/cm r/m/n/12 ), in-jek-tiv (gem[]a^^Y [] Overfull \hbox (28.64178pt too wide) in paragraph at lines 16355--16358 []\OT1/cmr/m/it/12 Beweis von Lem-ma 6: \OT1/cmr/m/n/12 Dies ist klar, denn das In-ver-se ei-nes $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulendomorphismus [] Overfull \hbox (84.23938pt too wide) in paragraph at lines 16355--16358 \OT1/cmr/m/n/12 ist wie-der ein $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodul endomorphismus, und das In-ver-se ei-nes $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -R echtscomodulhomomorphismus [] Overfull \hbox (26.53575pt too wide) in paragraph at lines 16363--16368 []\OT1/cmr/m/n/12 Nach Lem-ma 5 ist $^^H[] \OMS/cmsy/m/n/12 2 []$ \OT1/cmr/m/n/ 12 (denn $^^H[]$ ist das Bild von $[] \OMS/cmsy/m/n/12 2 [][]$ [] Overfull \hbox (1.55466pt too wide) in paragraph at lines 16372--16380 \OT1/cmr/m/n/12 ). We-gen $^^H[] = ^^H$ ist al-so $^^H \OMS/cmsy/m/n/12 2 []$\O T1/cmr/m/n/12 . So-mit ist auch $^^H[] \OMS/cmsy/m/n/12 2 []$ [] [209] Overfull \hbox (42.32065pt too wide) detected at line 16451 [] [] [210] Overfull \hbox (66.6178pt too wide) detected at line 16518 [] [] Overfull \hbox (0.32817pt too wide) in paragraph at lines 16518--16548 \OT1/cmr/m/n/12 Da $\OML/cmm/m/it/12 v[]; v[]; :::; v[]$ \OT1/cmr/m/n/12 ei-ne Ba-sis von $\OML/cmm/m/it/12 C[]$ \OT1/cmr/m/n/12 ist, lie-fert Ko-ef-fi-zi-en- ten-ver-gleich hieraus, da^^Y $^^A [] = [] [211] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 16568. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 16568. Overfull \hbox (80.9395pt too wide) in paragraph at lines 16617--16619 []\OT1/cmr/m/n/12 Nun k[]onnen wir al-ter-na-ti-ve De-fi-ni-tio-nen von $\OML/c mm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulalgebren und $\OML/cmm/m/it/12 H$\OT1/c mr/m/n/12 -Rechtscomodulalgebren [] [212] [213] Overfull \hbox (64.66444pt too wide) in paragraph at lines 16772--16779 \OT1/cmr/m/n/12 (wobei $\OML/cmm/m/it/12 x[] \OT1/cmr/m/n/12 = []$), dann wird $\OML/cmm/m/it/12 A \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 k []$ \OT1/cmr/m/n/12 au ch zu ei-ner $\OML/cmm/m/it/12 k [][]$\OT1/cmr/m/n/12 -Rechtscomodulalgebra [] Overfull \hbox (17.78868pt too wide) in paragraph at lines 16772--16779 \OT1/cmr/m/n/12 verm[]oge des Al-ge-bra-ho-mo-mor-phis-mus $[] : \OML/cmm/m/it/ 12 k [] \OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 k [] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 k [][]$\OT1/cmr/m/n/12 , [] [214] Overfull \hbox (37.85193pt too wide) in paragraph at lines 16825--16830 []\OT1/cmr/bx/n/12 b) \OT1/cmr/m/n/12 Sind $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 zwei $\OML/cmm/m/it/12 C$\OT1/cmr/m/n /12 -Rechtscomoduln, dann ist $\OMS/cmsy/m/n/12 M[] [] \OT1/cmr/m/n/12 = [][] [ ]$. [] Overfull \hbox (34.02733pt too wide) in paragraph at lines 16825--16830 \OT1/cmr/m/n/12 Hier-bei be-zeich-net $\OMS/cmsy/m/n/12 M[] []$ \OT1/cmr/m/n/12 (wie im-mer) die Men-ge der $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomodul homomorphismen [] Overfull \hbox (21.09923pt too wide) detected at line 16847 [] [] [215] [216] Overfull \hbox (12.20273pt too wide) in paragraph at lines 16981--16987 \OT1/cmr/m/n/12 die je-der $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulal gebrastruktur $\OML/cmm/m/it/12 ^^N$ \OT1/cmr/m/n/12 auf $\OML/cmm/m/it/12 A$ \ OT1/cmr/m/n/12 die zu $\OML/cmm/m/it/12 ^^N$ \OT1/cmr/m/n/12 ad-jun-gier-te $\O ML/cmm/m/it/12 H[]$\OT1/cmr/m/n/12 -Modulstruktur [] Overfull \hbox (40.30537pt too wide) in paragraph at lines 16981--16987 \OT1/cmr/m/n/12 die zu $\OML/cmm/m/it/12 ^^N$ \OT1/cmr/m/n/12 ad-jun-gier-te $\ OML/cmm/m/it/12 H[]$\OT1/cmr/m/n/12 -Modulstruktur $\OML/cmm/m/it/12 ^^V$ \OT1/ cmr/m/n/12 auch tats[]achlich ei-ne $\OML/cmm/m/it/12 H[]$\OT1/cmr/m/n/12 -Link smodulalgebrastruktur!) [] Overfull \hbox (12.31355pt too wide) detected at line 17050 [] [] [217] Overfull \hbox (14.78798pt too wide) detected at line 17074 [] [] [218] Overfull \hbox (31.67027pt too wide) in paragraph at lines 17128--17131 []\OT1/cmr/m/n/12 Wir ha-ben da-mit ge-zeigt: F[]ur je-des $\OML/cmm/m/it/12 G \OMS/cmsy/m/n/12 2 [] []$ \OT1/cmr/m/n/12 gibt es ein $\OML/cmm/m/it/12 F \OMS/ cmsy/m/n/12 2 [] []$ [] Overfull \hbox (27.16483pt too wide) in paragraph at lines 17212--17219 []\OT1/cmr/bx/n/12 b) \OT1/cmr/m/n/12 Sind $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 zwei $\OML/cmm/m/it/12 C[]$\OT1/cmr/m /n/12 -Linksmoduln, dann ist $[][] [] = \OMS/cmsy/m/n/12 M[] []$\OT1/cmr/m/n/12 . [] Overfull \hbox (86.12114pt too wide) in paragraph at lines 17212--17219 \OT1/cmr/m/n/12 Hier-bei be-zeich-net $\OMS/cmsy/m/n/12 M[] []$ \OT1/cmr/m/n/12 (wie im-mer) die Men-ge der $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomodul homomorphismen [] [219] Overfull \hbox (15.03986pt too wide) detected at line 17297 [] [] Overfull \hbox (44.95839pt too wide) detected at line 17335 [] [] [220] Overfull \hbox (13.21497pt too wide) detected at line 17405 [] [] [221] [222] Overfull \hbox (46.90929pt too wide) in paragraph at lines 17573--17575 []\OT1/cmr/m/n/12 Genau dann ist $[]$ ei-ne $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra, wenn $[]$ ei-ne $\OML/cmm/m/it/12 H[]$\OT1/cmr/m/n/12 - Linksmodulalgebra [] [223] Overfull \hbox (27.36256pt too wide) detected at line 17615 [] [] Overfull \hbox (3.74101pt too wide) detected at line 17630 [] [] Overfull \hbox (51.2245pt too wide) detected at line 17676 [] [] [224] Overfull \hbox (11.24664pt too wide) detected at line 17719 [] [] Overfull \hbox (20.30682pt too wide) detected at line 17749 [] [] [225] Overfull \hbox (2.68393pt too wide) in paragraph at lines 17779--17782 \OT1/cmr/m/n/12 die je-der $\OML/cmm/m/it/12 C[]$\OT1/cmr/m/n/12 -Linksmodulstr uktur $\OML/cmm/m/it/12 ^^V$ \OT1/cmr/m/n/12 auf $\OML/cmm/m/it/12 V$ \OT1/cmr/ m/n/12 die zu $\OML/cmm/m/it/12 ^^V$ \OT1/cmr/m/n/12 ad-jun-gier-te $\OML/cmm/m /it/12 C$\OT1/cmr/m/n/12 -Rechtscomodulstruktur [] Overfull \hbox (1.65523pt too wide) in paragraph at lines 17818--17828 []\OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 C$ \OT1/cmr/m/n/12 ei-ne Coal-ge-bra, s ei $\OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 2 \U/msb/m/n/12 N$\OT1/cmr/m/n/12 , und sei-en $\OML/cmm/m/it/12 V[]$\OT1/cmr/m/n/12 , $\OML/cmm/m/it/12 V[]$\OT1/cmr/ m/n/12 , $\OML/cmm/m/it/12 :::$\OT1/cmr/m/n/12 , $\OML/cmm/m/it/12 V[]$ \OT1/cm r/m/n/12 be-lie-bi-ge $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomoduln. [] [226] Overfull \hbox (37.03214pt too wide) detected at line 17940 [] [] [227] [228] Overfull \hbox (0.56065pt too wide) in paragraph at lines 17985--17992 []\OT1/cmr/m/n/12 Wir ha-ben wei-ter oben auf der di-rek-ten Sum-me $[] \OML/cm m/m/it/12 V[]$ \OT1/cmr/m/n/12 ei-ne $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Recht scomodulstruktur [] Overfull \hbox (10.90742pt too wide) in paragraph at lines 18005--18016 \OT1/cmr/m/n/12 iden-tisch sind, ist klar (denn als Vek-torr[]aume sind $[][] [ ] = [] \OML/cmm/m/it/12 V[]$ \OT1/cmr/m/n/12 und $[] [][] \OML/cmm/m/it/12 V[] \OT1/cmr/m/n/12 = [] Overfull \hbox (30.52359pt too wide) in paragraph at lines 18017--18022 []\OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 ^^V[]$ \OT1/cmr/m/n/12 die $\OML/cmm/m/ it/12 C[]$\OT1/cmr/m/n/12 -Linksmodulstruktur auf $[][] []$, und sei $\OML/cmm/ m/it/12 ^^V$ \OT1/cmr/m/n/12 die $\OML/cmm/m/it/12 C[]$\OT1/cmr/m/n/12 -Linksmo dulstruktur [] [229] [230] [231] [232] Overfull \hbox (35.61743pt too wide) in paragraph at lines 18266--18273 \OT1/cmr/m/n/12 sen-det je-des Ele-ment $\OML/cmm/m/it/12 a \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 F []$ \OT1/cmr/m/n/12 auf die nat[]urliche Trans-for-ma-ti-on $[][]$ [] Overfull \hbox (17.03606pt too wide) in paragraph at lines 18285--18292 \OT1/cmr/m/n/12 ist bi-jek-tiv. Da-bei ver-ste-hen wir un-ter $\OMS/cmsy/m/n/12 C []$ \OT1/cmr/m/n/12 die nat[]urliche Trans-for-ma-ti-on $[][] \OML/cmm/m/it/ 12 ;$ [] [233] Overfull \hbox (9.55177pt too wide) detected at line 18384 [] [] [234] [235] Underfull \hbox (badness 4391) in paragraph at lines 18531--18531 [][][]\OT1/cmr/m/it/10 Beweis: \OT1/cmr/m/n/10 Sie ist ei-ne Bi-jek-ti-on gem[] a^^Y der uni-ver-sel-len Ei-gen-schaft des Po-ly-nom-rings [] Overfull \hbox (42.75793pt too wide) in paragraph at lines 18558--18564 \OT1/cmr/m/n/12 von Men-gen, wo-bei $\OML/cmm/m/it/12 d \OT1/cmr/m/n/12 = [] [] $. (Da-bei be-zeich-nen wir mit $\OML/cmm/m/it/12 k []$ [] [236] [237] Overfull \hbox (23.1091pt too wide) detected at line 18707 \OT1/cmr/m/n/12 ^^H [] = ^^H [] = ^^H [] ^^H [] = [] [] [] [] = \OML/cmm/m/it/1 2 ' [] [] [] Overfull \hbox (28.24585pt too wide) detected at line 18756 \OML/cmm/m/it/12 G [] [] [] [] \OT1/cmr/m/n/12 = [] [] [] [] [238] [239] [240] [241] [242] Overfull \hbox (2.83316pt too wide) in paragraph at lines 19065--19068 \OT1/cmr/m/n/12 von nat[]urlichen Trans-for-ma-tio-nen zwi-schen Funk-to-ren ko m-mu-tiert, wo-bei die nat[]urliche [] [243] [244] [245] Overfull \hbox (33.61864pt too wide) detected at line 19239 [] [] [246] Overfull \hbox (0.63127pt too wide) in paragraph at lines 19325--19330 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Die Um-keh-rung von \OT1/cmr/bx/n/12 1) \ OT1/cmr/m/n/12 gilt nicht: Die ka-no-ni-sche In-klu-si-ons-ab-bil-dung $\OML/cm m/m/it/12 ' \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 k [] ,[]\OMS/cmsy/m/n/12 ! [] [247] [248] Overfull \hbox (1.56073pt too wide) detected at line 19427 [] [] Overfull \hbox (22.65831pt too wide) in paragraph at lines 19427--19434 \OT1/cmr/m/n/12 Nun ha-ben wir ka-no-ni-sche Iso-mor-phis-men $\U/msb/m/n/12 A[ ] [] [] [] []$ \OT1/cmr/m/n/12 und $[][] [] [] [] Overfull \hbox (2.79108pt too wide) detected at line 19455 [] [] Overfull \hbox (14.52527pt too wide) in paragraph at lines 19473--19478 \OT1/cmr/m/n/12 zwar die glei-che, wie die in Bei-spiel 4.5. \OT1/cmr/bx/n/12 5 ) \OT1/cmr/m/n/12 de-fi-nier-te $\OML/cmm/m/it/12 k [][]$\OT1/cmr/m/n/12 -Comod ulalgebrastruktur [] [249] [250] Overfull \hbox (8.33156pt too wide) detected at line 19580 [] [] Overfull \hbox (26.48308pt too wide) detected at line 19616 [] [] [251] [252] [253] Overfull \hbox (14.87247pt too wide) in paragraph at lines 19769--19771 []\OT1/cmr/m/n/12 Wir sind al-so in fol-gen-der Si-tua-ti-on: Wir ha-ben einen end-lich er-zeug-ten $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Rechtsmodul [] [254] [255] [256] [257] Overfull \hbox (15.24898pt too wide) in paragraph at lines 20135--20142 []\OT1/cmr/m/n/12 Anwendung von 6.2 \OT1/cmr/bx/n/12 c) \OT1/cmr/m/n/12 auf $\O ML/cmm/m/it/12 n \OT1/cmr/m/n/12 = 1 \OMS/cmsy/m/n/12 ^^@ [] \OML/cmm/m/it/12 q $ \OT1/cmr/m/n/12 er-gibt, da^^Y das Ele-ment $1 \OMS/cmsy/m/n/12 ^^@ []$ [] Overfull \hbox (15.24898pt too wide) in paragraph at lines 20143--20150 []\OT1/cmr/m/n/12 Anwendung von 6.2 \OT1/cmr/bx/n/12 c) \OT1/cmr/m/n/12 auf $\O ML/cmm/m/it/12 n \OT1/cmr/m/n/12 = 1 \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 q [] $ \OT1/cmr/m/n/12 er-gibt, da^^Y das Ele-ment $1 \OMS/cmsy/m/n/12 ^^@ []$ [] [258] [259] Overfull \hbox (21.76111pt too wide) in paragraph at lines 20309--20313 []\OT1/cmr/bx/n/12 6.4$[]$. Pro-po-si-ti-on: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it /12 R$ \OT1/cmr/m/n/12 ein ar-tin-scher ein-fa-cher Ring. Dann gibt es einen Sc hiefk[]orper [] [260] Overfull \hbox (25.41826pt too wide) detected at line 20418 [] [] [261] Overfull \hbox (39.02942pt too wide) detected at line 20504 [] [] [262] [263] [264] Overfull \hbox (2.56918pt too wide) in paragraph at lines 20690--20692 []\OT1/cmr/m/n/12 Dort wird die-ser Satz zu-wei-len in all-ge-mei-ne-ren Ver-si o-nen for-mu-liert; die-se ben[]otigen [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 20695. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 20695. Overfull \hbox (3.92642pt too wide) in paragraph at lines 20720--20726 []\OT1/cmr/bx/n/12 (a) \OT1/cmr/m/n/12 Un-ter ei-ner \OT1/cmr/m/it/12 vollst[]a ndigen Zer-le-gung \OT1/cmr/m/n/12 von $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ver -ste-hen wir ei-ne Lis-te $[]$ [] [265] Underfull \hbox (badness 10000) in paragraph at lines 20802--20802 \OT1/cmr/m/n/10 und $[]$ des $\OML/cmm/m/it/10 A$\OT1/cmr/m/n/10 -Linksmoduls $ \OML/cmm/m/it/10 V$ \OT1/cmr/m/n/10 not-wen-di-ger-wei-se $[][] [] = [] Overfull \hbox (48.99474pt too wide) in paragraph at lines 20731--20804 []\OT1/cmr/m/n/12 F[]ur je-de vollst[]andige Zer-le-gung $[]$ von $\OML/cmm/m/i t/12 V$ \OT1/cmr/m/n/12 be-zeich-nen wir mit $[][] []$ [] Overfull \hbox (30.91095pt too wide) in paragraph at lines 20731--20804 \OT1/cmr/m/n/12 die An-zahl al-ler $\OML/cmm/m/it/12 i \OMS/cmsy/m/n/12 2 []$\O T1/cmr/m/n/12 , wel-che $\OML/cmm/m/it/12 V[] [] E$ \OT1/cmr/m/n/12 erf[]ullen[ ][][]. Die-se An-zahl $[][] []$ [] Overfull \hbox (17.07462pt too wide) in paragraph at lines 20805--20808 []\OT1/cmr/m/n/12 Diese An-zahl $[] []$ hei^^Yt die \OT1/cmr/m/it/12 Viel-fach- heit \OT1/cmr/m/n/12 des $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmoduls $\OML /cmm/m/it/12 E$ \OT1/cmr/m/n/12 im $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmo dul [] [266] Overfull \hbox (21.79422pt too wide) in paragraph at lines 20828--20832 []\OT1/cmr/m/n/12 Sei $[]$ ei-ne vollst[]andige Zer-le-gung von $\OML/cmm/m/it/ 12 V$\OT1/cmr/m/n/12 . Dann ist $[]$ [] Overfull \hbox (33.01205pt too wide) detected at line 20855 [] [] [267]pdfTeX warning (ext4): destination with the same identifier (name{equation .1.1}) has been already used, duplicate ignored \relax l.20930 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.20930 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.20930 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.20930 \begin{equation} [268] Overfull \hbox (1.9013pt too wide) in paragraph at lines 21005--21010 \OT1/cmr/m/n/12 Wenden wir ge-nau das glei-che Ar-gu-ment auf den $\OML/cmm/m/i t/12 A$\OT1/cmr/m/n/12 -Linksmodul $\OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 mit der vollst[]andigen [] Overfull \hbox (8.70953pt too wide) in paragraph at lines 21045--21056 []\OT1/cmr/m/n/12 Wir de-fi-nie-ren f[]ur je-des $\OML/cmm/m/it/12 i \OMS/cmsy/ m/n/12 2 []$ \OT1/cmr/m/n/12 einen Un-ter-mo-dul $\OML/cmm/m/it/12 P[]$ \OT1/cm r/m/n/12 des $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmoduls [] Overfull \hbox (41.23659pt too wide) detected at line 21079 [] [] [269] Overfull \hbox (19.76134pt too wide) in paragraph at lines 21125--21130 \OT1/cmr/m/n/12 Teil-men-gen $[]$ und $[]$). [] Overfull \hbox (45.39238pt too wide) detected at line 21190 [] [] [270] Overfull \hbox (43.77524pt too wide) in paragraph at lines 21190--21195 []$ \OT1/cmr/m/n/12 an-stel-le des $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmo duls $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 mit der vollst[]andigen Zer-le-gung $ []$). [] Overfull \hbox (21.78096pt too wide) in paragraph at lines 21211--21215 []\OT1/cmr/bx/n/12 (a) \OT1/cmr/m/n/12 Sei-en $\OML/cmm/m/it/12 V[]$\OT1/cmr/m/ n/12 , $\OML/cmm/m/it/12 V[]$\OT1/cmr/m/n/12 , $\OML/cmm/m/it/12 :::$\OT1/cmr/m /n/12 , $\OML/cmm/m/it/12 V[]$ \OT1/cmr/m/n/12 end-lich-di-men-sio-na-le $\OML/ cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmoduln. Dann ist $[] [] = [] [271] [272] [273]pdfTeX warning (ext4): destination with the same identifier (n ame{equation.1.1}) has been already used, duplicate ignored \relax l.21468 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21468 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21468 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21468 \begin{equation} [274]pdfTeX warning (ext4): destination with the same identifier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21492 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21492 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21492 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21492 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21504 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21504 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21504 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21504 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21536 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21536 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21536 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.1.1}) has been already used, duplicate ignored \relax l.21536 \begin{equation} Overfull \hbox (8.41458pt too wide) in paragraph at lines 21545--21551 \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 ^^Z[]$ \OT1/cmr/m/n/12 sind $\OML/cmm/m/i t/12 k$\OT1/cmr/m/n/12 -Algebrahomomorphismen (denn $\OML/cmm/m/it/12 ^^Z[]$ \O T1/cmr/m/n/12 ist ein $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Algebrahomomorphismu s [] [275] [276] Overfull \hbox (6.34612pt too wide) in paragraph at lines 21743--21752 []\OT1/cmr/m/n/12 Da $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ein $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomodul ist, ist $[][] \OML/cmm/m/it/12 V$ \OT1/cmr/m /n/12 ein $\OML/cmm/m/it/12 C[]$\OT1/cmr/m/n/12 -Linksmodul, al-so ein $\OML/cm m/m/it/12 B$\OT1/cmr/m/n/12 -Linksmodul. [] [277] Overfull \hbox (42.37685pt too wide) in paragraph at lines 21775--21787 \OML/cmm/m/it/12 V[]$ \OT1/cmr/m/n/12 nach $\OML/cmm/m/it/12 W[]$\OT1/cmr/m/n/1 2 . Da $^^H$ ein Vek-tor-rau-mi-so-mor-phis-mus ist, ist al-so $^^H$ ein $\OML/ cmm/m/it/12 B$\OT1/cmr/m/n/12 -Linksmodulisomorphismus [] Overfull \hbox (22.39667pt too wide) detected at line 21795 [] [] Overfull \hbox (71.01828pt too wide) in paragraph at lines 21795--21800 \OT1/cmr/m/n/12 Damit wird $\OML/cmm/m/it/12 ^^^ \OMS/cmsy/m/n/12 2 [][] []$ \O T1/cmr/m/n/12 zu $\OML/cmm/m/it/12 ^^^ \OMS/cmsy/m/n/12 2 M[] []$\OT1/cmr/m/n/1 2 . So-mit ist $\OML/cmm/m/it/12 ^^^$ \OT1/cmr/m/n/12 ein $\OML/cmm/m/it/12 C$\ OT1/cmr/m/n/12 -Rechtscomodulhomomorphismus. [] Overfull \hbox (47.57738pt too wide) in paragraph at lines 21795--21800 \OT1/cmr/m/n/12 Da $\OML/cmm/m/it/12 ^^^$ \OT1/cmr/m/n/12 au-^^Yer-dem ein Vek- tor-rau-mi-so-mor-phis-mus ist, ist al-so $\OML/cmm/m/it/12 ^^^$ \OT1/cmr/m/n/1 2 ein $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtscomodulisomorphismus. [] Overfull \hbox (21.18913pt too wide) in paragraph at lines 21813--21813 []\OT1/cmr/bx/n/14.4 Abschnitt 1. Lie-al-ge-bren und ih-re uni-ver-sel-len Einh []ullenden [] [278] [279] [280] Overfull \hbox (20.09917pt too wide) detected at line 22056 [] [] Overfull \hbox (33.24014pt too wide) detected at line 22105 [] [] [281] Overfull \hbox (30.25977pt too wide) detected at line 22170 [] [] [282] Overfull \hbox (0.9055pt too wide) in paragraph at lines 22207--22214 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/m/n/12 F[]ur al-le $\OML/cmm/m/it/12 D; E \ OMS/cmsy/m/n/12 2 [] [][]$ \OT1/cmr/m/n/12 und al-le $\OML/cmm/m/it/12 ^^U \OMS /cmsy/m/n/12 2 \OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 D \OT1 /cmr/m/n/12 + \OML/cmm/m/it/12 E \OMS/cmsy/m/n/12 2 [] [][]$\OT1/cmr/m/n/12 , [] Overfull \hbox (76.23358pt too wide) detected at line 22285 [] [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 " [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/1 2 " [] " [] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 " [] " [] \OT1/cmr/m/n/12 = \OML /cmm/m/it/12 " [] [] [] \OT1/cmr/m/n/12 + [] [] \OML/cmm/m/it/12 " [] ; [] Overfull \hbox (56.94113pt too wide) detected at line 22308 [] [] [283] Overfull \hbox (21.98634pt too wide) detected at line 22326 [] [] [284] Overfull \hbox (3.34123pt too wide) in paragraph at lines 22436--22440 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 Q \OMS/cmsy/m/n/12 2 [][] []$ \OT1/cmr/m/n/12 ei-ne Ma-trix. Dann ist $\U/euf/m/n/12 o [] \OT1/cmr /m/n/12 := [] \OMS/cmsy/m/n/12 ^^R [] Overfull \hbox (1.04248pt too wide) in paragraph at lines 22517--22524 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 F[]ur je-des $\OML/cmm/m/it/12 d \OMS/cms y/m/n/12 2 [][] []$ \OT1/cmr/m/n/12 exis-tiert ge-nau ei-ne $\OML/cmm/m/it/12 k $\OT1/cmr/m/n/12 -lineare Ab-bil-dung von $\OML/cmm/m/it/12 A[]\U/msb/m/n/12 ^^ ^ [][]$ [] [285] Overfull \hbox (60.15654pt too wide) detected at line 22662 [] [] [286] [287] Overfull \hbox (3.76369pt too wide) in paragraph at lines 22750--22754 \OT1/cmr/m/n/12 (wobei $\OMS/cmsy/m/n/12 ^^C$ \OT1/cmr/m/n/12 die Mul-ti-pli-ka -ti-on in $\OML/cmm/m/it/12 G []$\OT1/cmr/m/n/12 , al-so die Fal-tung in $\OML/ cmm/m/it/12 G [] \OT1/cmr/m/n/12 = [] []$ [] [288] Overfull \hbox (67.77571pt too wide) detected at line 22844 [] [] Overfull \hbox (39.82591pt too wide) in paragraph at lines 22844--22849 \OT1/cmr/m/n/12 ), al-so ein Ele-ment von $\OML/cmm/m/it/12 G []$\OT1/cmr/m/n/1 2 . Fer-ner erf[]ullt die-se Ab-bil-dung die Glei-chung $[] [] = [] Overfull \hbox (30.49467pt too wide) in paragraph at lines 22873--22883 \OT1/cmr/m/n/12 jek-tiv ist. Da-zu sei $\OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 2 [ ] \OML/cmm/m/it/12 G$ \OT1/cmr/m/n/12 ein be-lie-bi-ges Ele-ment. We-gen $[] \O ML/cmm/m/it/12 G \OT1/cmr/m/n/12 = [] []$ [] [289] Overfull \hbox (44.45663pt too wide) in paragraph at lines 22884--22891 []\OT1/cmr/m/n/12 Sei nun $\OML/cmm/m/it/12 ^^Y[] \OT1/cmr/m/n/12 : \OML/cmm/m/ it/12 k [] \OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 der $\OML/cmm /m/it/12 k$\OT1/cmr/m/n/12 -Vektorraumhomomorphismus (kein $\OML/cmm/m/it/12 k$ \OT1/cmr/m/n/12 -Algebrahomomorphismus), [] Overfull \hbox (56.14716pt too wide) detected at line 22918 [] [] Overfull \hbox (12.56635pt too wide) detected at line 22952 [] [] [290] [291] Overfull \hbox (74.73991pt too wide) detected at line 23091 [] [] Overfull \hbox (61.66779pt too wide) detected at line 23232 [] [] [292] [293] Overfull \hbox (57.47292pt too wide) detected at line 23314 [] [] [294] [295] [296] Overfull \hbox (59.51318pt too wide) detected at line 23563 [] [] Overfull \hbox (50.57011pt too wide) in paragraph at lines 23563--23571 \OT1/cmr/m/n/12 af-fi-nen Grup-pe $\OML/cmm/m/it/12 G$ \OT1/cmr/m/n/12 ar-bei-t en wol-len, k[]onnen wir in je-dem ih-rer ``Avat-a-re'' $[] []$, [] Overfull \hbox (57.37495pt too wide) detected at line 23627 [] [] [297] Overfull \hbox (38.67906pt too wide) detected at line 23668 [] [] [298] [299] Overfull \hbox (69.38078pt too wide) detected at line 23898 [] [] [300] [301] Overfull \hbox (19.27777pt too wide) detected at line 24004 [] [] Underfull \hbox (badness 1371) in paragraph at lines 24047--24047 [][][]\OT1/cmr/m/n/10 So ein $\OML/cmm/m/it/10 U []$ \OT1/cmr/m/n/10 exis-tiert und ist ein-deu-tig. Um dies ein-zu-se-hen, m[]ussen wir nur 1.9. \OT1/cmr/bx/ n/10 2) [] Underfull \hbox (badness 1189) in paragraph at lines 24047--24047 \OT1/cmr/m/n/10 auf $\U/euf/m/n/10 g[]$\OT1/cmr/m/n/10 , $\OML/cmm/m/it/10 U [] $ \OT1/cmr/m/n/10 und $[]$ statt $\U/euf/m/n/10 g$\OT1/cmr/m/n/10 , $\OML/cmm/m /it/10 A$ \OT1/cmr/m/n/10 und $\OML/cmm/m/it/10 f$ \OT1/cmr/m/n/10 an-wen-den ( denn [] [302] Overfull \hbox (21.06572pt too wide) in paragraph at lines 24083--24089 \OT1/cmr/m/n/12 Dann hei^^Yt $\OML/cmm/m/it/12 f$ \OT1/cmr/m/n/12 ein \OT1/cmr/ m/it/12 Ho-mo-mor-phis-mus von $\U/euf/m/n/12 g$\OT1/cmr/m/it/12 -Moduln \OT1/c mr/m/n/12 (oder auch ein $\U/euf/m/n/12 g$\OT1/cmr/m/it/12 -Modulhomomorphismus \OT1/cmr/m/n/12 , [] Underfull \hbox (badness 3547) in paragraph at lines 24117--24117 [][][]\OT1/cmr/m/n/10 So ei-ne $\OML/cmm/m/it/10 U []$\OT1/cmr/m/n/10 -Modulstr uktur exis-tiert und ist ein-deu-tig, denn wenn wir sie als Al- [] Underfull \hbox (badness 3281) in paragraph at lines 24117--24117 \OT1/cmr/m/n/10 ge-bra-ho-mo-mor-phis-mus $\OML/cmm/m/it/10 ' \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 U [] \OMS/cmsy/m/n/10 ! [] \OML/cmm/m/it/10 V$ \OT1/cmr/m/n/1 0 be-trach-ten, ver-ein-facht sich die Be-din-gung [] Overfull \hbox (32.33742pt too wide) detected at line 24129 [] [] [303] Overfull \hbox (29.69969pt too wide) detected at line 24216 [] [] [304] Underfull \hbox (badness 10000) in paragraph at lines 24280--24280 [][][]\OT1/cmr/m/n/10 und zwar fol-gen-der-ma-^^Yen: F[]ur je-des $\OML/cmm/m/i t/10 k \OMS/cmsy/m/n/10 2 \U/msb/m/n/10 N$ \OT1/cmr/m/n/10 ist $\OML/cmm/m/it/1 0 U[] []$ \OT1/cmr/m/n/10 der Un-ter-vek-tor-raum [] [305] Overfull \hbox (59.81004pt too wide) in paragraph at lines 24320--24330 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 F[]ur al-le $\OML/cmm/m/it/12 n \OMS/cmsy /m/n/12 ^^U \OT1/cmr/m/n/12 0$ wird der Vek-tor-raum $\OML/cmm/m/it/12 U[] [] \ U/msb/m/n/12 ^^^\OML/cmm/m/it/12 U[] []$ \OT1/cmr/m/n/12 er-zeugt von al-len $[ ]$ [] Overfull \hbox (9.82535pt too wide) detected at line 24405 [] [] [306] Overfull \hbox (5.24364pt too wide) detected at line 24526 [] \OML/cmm/m/it/12 : [] [307] [308] Overfull \hbox (0.61743pt too wide) in paragraph at lines 24636--24656 []$ \OT1/cmr/m/n/12 ist li-ne-ar un-abh[]angig[][][]. Da [] [309] Overfull \hbox (1.58008pt too wide) in paragraph at lines 24714--24720 []\OT1/cmr/m/it/12 Induktionsschritt: \OT1/cmr/m/n/12 An-ge-nom-men, wir ha-ben be-reits ei-ne $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -bilineare Ab-bil-dung $\OM L/cmm/m/it/12 ^^V[] \OT1/cmr/m/n/12 : [] Overfull \hbox (39.87013pt too wide) detected at line 24834 [] [] [310] Overfull \hbox (12.05153pt too wide) detected at line 24930 [] [] [311] Overfull \hbox (33.18419pt too wide) detected at line 24959 [] [] [312] Overfull \hbox (48.76846pt too wide) detected at line 25025 [] [] [313] [314] Overfull \hbox (39.93584pt too wide) detected at line 25195 [] [] Overfull \hbox (1.01079pt too wide) in paragraph at lines 25225--25228 []\OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 I \OMS/cmsy/m/n/12 ^^R \OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ein Recht-s-ide-al. Dann ist $[][]$ ei-ne Fil-trie-rung des $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmoduls [] [315] Overfull \hbox (12.5796pt too wide) in paragraph at lines 25250--25274 \OML/cmm/m/it/12 n \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 1$ auf $\OML/cmm/m/it/1 2 n$\OT1/cmr/m/n/12 ) sei $\OML/cmm/m/it/12 x \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/ 12 I \OMS/cmsy/m/n/12 \ \OML/cmm/m/it/12 A[]:$ \OT1/cmr/m/n/12 Wir be-trach-ten die []Aquivalenzklasse $[] \OMS/cmsy/m/n/12 2 [] \U/msb/m/n/12 ^^^ [] \OML/cmm /m/it/12 :$ [] [316] Overfull \hbox (25.3822pt too wide) in paragraph at lines 25314--25317 []\OT1/cmr/m/n/12 Diese Fa-mi-lie $[][]$ hei^^Yt ei-ne \OT1/cmr/m/it/12 Coal-ge -bra-fil-trie-rung \OT1/cmr/m/n/12 der Coal-ge-bra $\OML/cmm/m/it/12 C$\OT1/cmr /m/n/12 , wenn $[]$ [] [317] Overfull \hbox (83.13066pt too wide) detected at line 25426 [] [] Overfull \hbox (23.67122pt too wide) detected at line 25459 [] [] \OML/cmm/m/it/12 ::: [] \OT1/cmr/m/n/12 = [] [] \OML/cmm/m/it/12 a[]a[]:: :a[] \OMS/cmsy/m/n/12 ^^A \OML/cmm/m/it/12 b[]b[]:::b[]: [] [318] [319] Overfull \hbox (21.36705pt too wide) detected at line 25654 [] [] [320] [321] Overfull \hbox (19.4079pt too wide) in paragraph at lines 25700--25708 []\OT1/cmr/m/n/12 F[]ur je-des $\OML/cmm/m/it/12 a \OMS/cmsy/m/n/12 2 \U/msb/m/ n/12 N[]$ \OT1/cmr/m/n/12 de-fi-nie-ren wir ein Ele-ment $\OML/cmm/m/it/12 e[] \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 U []$ \OT1/cmr/m/n/12 durch $\OML/cmm/m/it/ 12 e[] \OT1/cmr/m/n/12 = []$, [] Overfull \hbox (23.5401pt too wide) in paragraph at lines 25700--25708 \OT1/cmr/m/n/12 wo-bei die Ele-men-te $\OML/cmm/m/it/12 i[]; i[]; :::; i[] \OMS /cmsy/m/n/12 2 \OML/cmm/m/it/12 I$ \OT1/cmr/m/n/12 durch die Be-din-gun-gen $[] = []$ [] Overfull \hbox (15.49799pt too wide) in paragraph at lines 25781--25786 []\OT1/cmr/m/n/12 Wir de-fi-nie-ren ei-ne Ab-bil-dung $\U/euf/m/n/12 b \OT1/cmr /m/n/12 : \U/euf/m/n/12 S \OMS/cmsy/m/n/12 ! \U/msb/m/n/12 N[]$ \OT1/cmr/m/n/12 fol-gen-der-ma-^^Yen: F[]ur je-des $[] \OMS/cmsy/m/n/12 2 [] Overfull \hbox (14.01361pt too wide) in paragraph at lines 25799--25804 []\OT1/cmr/m/n/12 Wir de-fi-nie-ren ei-ne Ab-bil-dung $\U/euf/m/n/12 c \OT1/cmr /m/n/12 : \U/euf/m/n/12 S \OMS/cmsy/m/n/12 ! \U/msb/m/n/12 N[]$ \OT1/cmr/m/n/12 fol-gen-der-ma-^^Yen: F[]ur je-des $[] \OMS/cmsy/m/n/12 2 [] [322] Overfull \hbox (9.89322pt too wide) detected at line 25875 [] [] Overfull \hbox (2.43613pt too wide) detected at line 25894 [] [] Overfull \hbox (18.24149pt too wide) detected at line 25946 [] [] [323] [324] [325] Overfull \hbox (56.25644pt too wide) in paragraph at lines 26096--26100 []\OT1/cmr/m/n/12 Diese Ab-bil-dung ist ein Al-ge-bra-ho-mo-mor-phis-mus nach \ OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 und ein $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Vektorraumisomorphismus [] Overfull \hbox (1.4021pt too wide) in paragraph at lines 26106--26114 []\OT1/cmr/m/n/12 In der Tat ist $^^A [] = \OML/cmm/m/it/12 x \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 1 + 1 \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 x$\OT1/cmr/m/n/12 . We-gen $\OML/cmm/m/it/12 x \OT1/cmr/m/n/1 2 = [] \OML/cmm/m/it/12 r[]e[]$ \OT1/cmr/m/n/12 wird dies zu $[] \OML/cmm/m/it/ 12 r[]\OT1/cmr/m/n/12 ^^A [] = [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 26140. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 26140. [326] Overfull \hbox (95.61458pt too wide) detected at line 26202 [] [] [327] Overfull \hbox (22.34724pt too wide) detected at line 26299 \OML/cmm/m/it/12 ' \OT1/cmr/m/n/12 = [] [] = [] [] = [] [] = [] [] = [] \OML/cm m/m/it/12 : [] [328] [329] Overfull \hbox (17.61594pt too wide) detected at line 26404 \OT1/cmr/m/n/12 ^^H [] [] = [] \OML/cmm/m/it/12 f [] c[]d[] [] [] [330] [331] Overfull \hbox (28.87584pt too wide) in paragraph at lines 26557--26573 []\OT1/cmr/bx/n/12 c) \OT1/cmr/m/n/12 Jetzt wer-den wir zei-gen, da^^Y $[] \OMS /cmsy/m/n/12 ^^R \OML/cmm/m/it/12 G []$ [] [332] Overfull \hbox (6.16579pt too wide) in paragraph at lines 26636--26647 \OT1/cmr/m/n/12 zu zei-gen, da^^Y sie ei-ne Bi-jek-ti-on ist, m[]ussen wir so-g ar zei-gen, da^^Y $[] [] = [] [333] Overfull \hbox (17.90422pt too wide) detected at line 26721 [] [] Overfull \hbox (54.4273pt too wide) detected at line 26729 [] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 P [] \OT1/cmr/m/n/12 = [] [] = [] \OML/cmm/m/it/12 : [] Overfull \hbox (6.70738pt too wide) detected at line 26758 [] [] [334] Overfull \hbox (27.92274pt too wide) in paragraph at lines 26816--26826 \OT1/cmr/m/n/12 ist kom-mu-ta-tiv. Da $\OML/cmm/m/it/12 P []$ \OT1/cmr/m/n/12 b i-jek-tiv ist, folgt hieraus, da^^Y die Ab-bil-dung $[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 G [] \OMS/cmsy/m/n/12 ! [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 26830. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 26830. [335] Overfull \hbox (8.76988pt too wide) detected at line 26915 [] [] [336] [337] Overfull \hbox (3.39777pt too wide) in paragraph at lines 27006--27009 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 D$ \OT1/cmr/m/n/12 ei-ne ir-re-du-zi-ble Un-ter-coal-ge-bra von $\OML/cmm/m/it/12 C:$ \OT1/cmr/m/n /12 Wir m[]ussen nur zei-gen: $[] \OML/cmm/m/it/12 C[]$ [] [338] Overfull \hbox (53.37302pt too wide) detected at line 27138 [] [] [] [] Overfull \hbox (22.84439pt too wide) detected at line 27144 [] [] [] \OML/cmm/m/it/12 : [] [339] Overfull \hbox (53.37302pt too wide) detected at line 27176 [] [] [] [] Overfull \hbox (22.84439pt too wide) detected at line 27182 [] [] [] \OML/cmm/m/it/12 : [] [340] Overfull \hbox (53.37302pt too wide) detected at line 27261 [] [] [] [] Overfull \hbox (22.84439pt too wide) detected at line 27267 [] [] [] \OML/cmm/m/it/12 : [] Overfull \hbox (38.54861pt too wide) detected at line 27276 [] [] [341] [342] Overfull \vbox (0.75723pt too high) has occurred while \output is active [] [343] Overfull \hbox (7.60165pt too wide) in paragraph at lines 27465--27473 []\OT1/cmr/m/n/12 Da je-de ein-fa-che Un-ter-coal-ge-bra von $\OML/cmm/m/it/12 D$ \OT1/cmr/m/n/12 in $\OML/cmm/m/it/12 D[]$ \OT1/cmr/m/n/12 ent-hal-ten ist (d enn $\OML/cmm/m/it/12 D[] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 F$\OT1/cmr/m/n/ 12 ), [] Overfull \hbox (93.58192pt too wide) in paragraph at lines 27474--27482 []\OT1/cmr/bx/n/12 4.10. Be-mer-kung: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 G$ \OT1/cmr/m/n/12 ei-ne Grup-pe. Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne H opfal-ge-bra. Ei-ne $\OML/cmm/m/it/12 k []$\OT1/cmr/m/n/12 -Linksmodulalgebrast ruktur [] Overfull \hbox (75.46841pt too wide) in paragraph at lines 27474--27482 \OT1/cmr/m/n/12 Hopfal-ge-bra-au-to-mor-phis-men von $\OML/cmm/m/it/12 H$ \OT1/ cmr/m/n/12 be-zeich-nen). Aus-ge-hend von der $\OML/cmm/m/it/12 k []$\OT1/cmr/m /n/12 -Linksmodulalgebrastruktur [] Overfull \hbox (43.6166pt too wide) in paragraph at lines 27512--27514 []\OT1/cmr/m/n/12 Dieser Un-ter-raum $\OML/cmm/m/it/12 H[]$ \OT1/cmr/m/n/12 ist au-^^Yer-dem ei-ne Un-ter-al-ge-bra von $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 , und ei-ne $\OML/cmm/m/it/12 G$\OT1/cmr/m/n/12 -Linksmodulalgebra. [] [344] [345] Overfull \hbox (3.15427pt too wide) in paragraph at lines 27658--27658 []\OT1/cmr/bx/n/14.4 Abschnitt 5. Hilfss[]atze []uber die dua-le Al-ge-bra ei-n er Coal- [] [346]pdfTeX warning (ext4): destination with the same identifier (name{equation .2.1}) has been already used, duplicate ignored \relax l.27738 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27738 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27738 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27738 \begin{equation} % [347] Overfull \hbox (1.99434pt too wide) detected at line 27827 \OML/cmm/m/it/12 b[] [] \OT1/cmr/m/n/12 = [] \OMS/cmsy/m/n/12 ^^@ [] \OT1/cmr/m /n/12 = \OML/cmm/m/it/12 u[] [] \OMS/cmsy/m/n/12 ^^@ [] [] \OT1/cmr/m/n/12 = \O ML/cmm/m/it/12 u[] [] \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 u[] [] \OT1/cmr/m/n /12 = 0\OML/cmm/m/it/12 ; [] pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1} ) has been already used, duplicate ignored \relax l.27902 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27902 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27902 \begin{equation} %pdfTeX warning (ext4): destination with the same ident ifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.27902 \begin{equation} % [348] Overfull \hbox (35.71645pt too wide) detected at line 27975 [] [] \OT1/cmr/m/n/12 = [] [] = [] [] + [] [] \OML/cmm/m/it/12 : [] Overfull \hbox (29.44244pt too wide) detected at line 27992 [] [] Overfull \hbox (49.8673pt too wide) detected at line 28004 [] [] \OT1/cmr/m/n/12 = [] [] = [] [] \OMS/cmsy/m/n/12 \ [] [] \OML/cmm/m/it/12 : [] [349] [350] Overfull \hbox (5.89168pt too wide) detected at line 28189 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1} ) has been already used, duplicate ignored \relax l.28219 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28219 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28219 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28219 \begin{equation} [351]pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28227 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28227 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28227 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28227 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28297 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28297 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28297 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28297 \begin{equation} [352]pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28307 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28307 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28307 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28307 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28334 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28334 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28334 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28334 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28341 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28341 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28341 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28341 \begin{equation} [353]pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28374 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28374 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28374 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.28374 \begin{equation} Overfull \hbox (42.89697pt too wide) detected at line 28411 [] [] Overfull \hbox (14.1785pt too wide) detected at line 28439 [] [] [354] [355] Overfull \hbox (5.00267pt too wide) detected at line 28552 \OML/cmm/m/it/10 f [] \OT1/cmr/m/n/10 = [] [] = \OML/cmm/m/it/10 g [] \OT1/cmr/ m/n/10 = 0 [] [] [356] [357] Overfull \hbox (21.69841pt too wide) detected at line 28715 \OT1/cmr/m/n/12 ^^@ [] = [] [] = [] [][] = [] \OML/cmm/m/it/12 I[] \OMS/cmsy/m/ n/12 \OML/cmm/m/it/12 J[]: [] [358] Overfull \hbox (53.61028pt too wide) detected at line 28817 [] [] Overfull \hbox (36.75192pt too wide) detected at line 28827 [] [] [359] [360] Overfull \hbox (44.68509pt too wide) detected at line 28958 [] [] Overfull \hbox (59.34035pt too wide) detected at line 28987 [] [] [361] Overfull \hbox (60.1164pt too wide) detected at line 29090 [] [] Overfull \hbox (24.6533pt too wide) detected at line 29110 [] [] [362] Overfull \hbox (0.33398pt too wide) in paragraph at lines 29149--29153 []\OT1/cmr/bx/n/12 Definition: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 A$ \OT1/cm r/m/n/12 ei-ne Al-ge-bra, und sei $[][]$ ei-ne Fa-mi-lie von Un-ter-vek-torr[]a umen [] [363]pdfTeX warning (ext4): destination with the same identifier (name{equation .2.1}) has been already used, duplicate ignored \relax l.29221 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29221 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29221 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29221 \begin{equation} Overfull \hbox (8.30182pt too wide) in paragraph at lines 29255--29265 []$ \OT1/cmr/m/n/12 folgt $\OML/cmm/m/it/12 u [] \OT1/cmr/m/n/12 = 0$ f[]ur al- le $\OML/cmm/m/it/12 c \OMS/cmsy/m/n/12 2 []$\OT1/cmr/m/n/12 , al-so $\OML/cmm/ m/it/12 u [] \OT1/cmr/m/n/12 = [] [364] Overfull \hbox (54.64432pt too wide) detected at line 29300 [] [] Overfull \hbox (73.61775pt too wide) in paragraph at lines 29307--29314 [] \OMS/cmsy/m/n/12 ^^R \OML/cmm/m/it/12 I[]$ \OT1/cmr/m/n/12 (da $\OML/cmm/m/i t/12 I[]$ \OT1/cmr/m/n/12 ein Vek-tor-raum ist). Da $[] = [] Overfull \hbox (38.03098pt too wide) detected at line 29330 [] [] [365] [366]pdfTeX warning (ext4): destination with the same identifier (name{eq uation.2.1}) has been already used, duplicate ignored \relax l.29404 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29404 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29404 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29404 \begin{equation} Overfull \hbox (33.78685pt too wide) detected at line 29422 [] [] pdfTeX warning (ext4): destination with the same identifier (name{equation.2.1} ) has been already used, duplicate ignored \relax l.29426 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29426 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29426 \begin{equation} pdfTeX warning (ext4): destination with the same identi fier (name{equation.2.1}) has been already used, duplicate ignored \relax l.29426 \begin{equation} [367] Overfull \hbox (1.49643pt too wide) detected at line 29528 [] [] [368] Overfull \hbox (0.90428pt too wide) detected at line 29564 \OML/cmm/m/it/12 C[] \OT1/cmr/m/n/12 = [] = [] \OML/cmm/m/it/12 D [] [] Overfull \hbox (30.3831pt too wide) in paragraph at lines 29589--29594 \OT1/cmr/m/n/12 nun aber $\OML/cmm/m/it/12 C[] \OT1/cmr/m/n/12 = ^^A[] []$, und ana-log ist $\OML/cmm/m/it/12 C[] \OT1/cmr/m/n/12 = ^^A[] []$. [] [369] [370] Overfull \hbox (16.63675pt too wide) detected at line 29709 [] [] [371] Overfull \hbox (30.38405pt too wide) detected at line 29804 [] [] Overfull \hbox (32.80614pt too wide) in paragraph at lines 29847--29853 []\OT1/cmr/m/n/12 Sei nun $\OML/cmm/m/it/12 c \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/ 12 C[]$ \OT1/cmr/m/n/12 be-lie-big. Nach der De-fi-ni-ti-on von $\OML/cmm/m/it/ 12 C[]$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 C[] \OT1/cmr/m/n/12 = ^^A[] []$. [] [372] Overfull \hbox (3.66776pt too wide) detected at line 29934 [] [] [373] Overfull \hbox (70.11398pt too wide) detected at line 29958 [] [] [374] [375] Overfull \hbox (3.72661pt too wide) in paragraph at lines 30151--30156 []\OT1/cmr/m/n/12 Idee: Sei $\OML/cmm/m/it/12 H[] \OMS/cmsy/m/n/12 ^^R \OML/cmm /m/it/12 H[] \OMS/cmsy/m/n/12 ^^R \OML/cmm/m/it/12 H[] \OMS/cmsy/m/n/12 ^^R \OM L/cmm/m/it/12 :::$ \OT1/cmr/m/n/12 die Co-ra-di-kal-fil-trie-rung. Dann ist $[] \OML/cmm/m/it/12 H \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 H[]\U/msb/m/n/12 ^^^\ OML/cmm/m/it/12 H[]$ [] [376] Overfull \hbox (2.8365pt too wide) in paragraph at lines 30168--30168 []\OT1/cmr/bx/n/17.28 III. Ka-pi-tel: End-lich-di-men-sio-na-le Hopfal-ge-bren [] Overfull \hbox (12.01897pt too wide) in paragraph at lines 30188--30201 []\OT1/cmr/m/it/12 Bedingung $\OMS/cmsy/m/n/12 B[]$\OT1/cmr/m/it/12 : \OT1/cmr/ m/n/12 Die Ab-bil-dung $\OML/cmm/m/it/12 ^^N[]$ \OT1/cmr/m/n/12 ist $\OML/cmm/m /it/12 H$\OT1/cmr/m/n/12 -rechtslinear, wo-bei die $\OML/cmm/m/it/12 H$\OT1/cmr /m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (60.49632pt too wide) in paragraph at lines 30220--30225 []\OT1/cmr/m/n/12 Ein $[]$\OT1/cmr/m/it/12 -Hopfmodul \OT1/cmr/m/n/12 ist dann de-fi-niert als ein Vek-tor-raum $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 mit ei-ne r $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (6.7472pt too wide) in paragraph at lines 30226--30233 []\OT1/cmr/m/it/12 Bedingung $\OMS/cmsy/m/n/12 C[]$\OT1/cmr/m/it/12 : \OT1/cmr/ m/n/12 Die Ab-bil-dung $\OML/cmm/m/it/12 ^^N[]$ \OT1/cmr/m/n/12 ist $\OML/cmm/m /it/12 A$\OT1/cmr/m/n/12 -rechtslinear, wo-bei die $\OML/cmm/m/it/12 A$\OT1/cmr /m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (14.23045pt too wide) in paragraph at lines 30226--30233 \OT1/cmr/m/n/12 auf-ge-fasst wird (weil $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ei n $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodul ist), und aus die-ser $\OML/ cmm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmodulstruktur [] [377] Overfull \hbox (2.21236pt too wide) in paragraph at lines 30310--30314 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/bx/n/12 1) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/ it/12 W \OMS/cmsy/m/n/12 2 M[]\OML/cmm/m/it/12 :$ \OT1/cmr/m/n/12 Wir m[]ussen zei-gen, da^^Y die Ab-bil-dung $\OML/cmm/m/it/12 W \OMS/cmsy/m/n/12 ! [][] \OML /cmm/m/it/12 ;$ [] [378] Overfull \hbox (18.05957pt too wide) detected at line 30411 [] [] [379] Overfull \hbox (32.95898pt too wide) in paragraph at lines 30497--30501 []\OT1/cmr/m/n/12 Die $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulstruktu r $\OML/cmm/m/it/12 ^^N[] \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 H[] \OMS/cmsy/m/n/ 12 ! \OML/cmm/m/it/12 H[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 auf $\OML/cmm/m/it/12 H[]$ \OT1/cmr/m/n/12 und die $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (85.96326pt too wide) detected at line 30539 [] [] [380] Overfull \hbox (1.79887pt too wide) in paragraph at lines 30588--30593 \OT1/cmr/m/n/12 und gem[]a^^Y der []Aquivalenzumformung folgt hieraus $\OML/cmm /m/it/12 ^^N[] [] \OT1/cmr/m/n/12 = [] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 p[]h[];$ [] [381] Overfull \hbox (3.62218pt too wide) detected at line 30687 [] [] [382] [383] Overfull \hbox (46.94212pt too wide) detected at line 30747 \OT1/cmr/m/n/12 ^^C\OML/cmm/m/it/12 g \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 g[] [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 g[]qgx[] \OT1/cmr/m/n/12 = \OML/cmm/m /it/12 q [] g[]x[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 q [] x[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 q [] g[]x[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 q\OT1/cmr/m/n /12 ^^C \OMS/cmsy/m/n/12 6\OT1/cmr/m/n/12 = ^^C\OML/cmm/m/it/12 : [] [384] Overfull \hbox (8.49423pt too wide) in paragraph at lines 30888--30895 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 Be-trach-ten wir erst-mal den Fall, wenn $\OML/cmm/m/it/12 ^^U \OMS/cmsy/m/n/12 2 [][] []$ \OT1/cmr/m/n/12 ist. Da $\OML /cmm/m/it/12 ^^U \OMS/cmsy/m/n/12 6\OT1/cmr/m/n/12 = 0$ und $[] [] = [] [385] Overfull \hbox (72.46458pt too wide) detected at line 30989 [] [] [386] Overfull \hbox (71.3076pt too wide) detected at line 31063 [] [] Overfull \hbox (61.26523pt too wide) detected at line 31096 [] [] [387] Overfull \hbox (36.07402pt too wide) detected at line 31161 [] [] [388] [389] [390] Overfull \hbox (6.27007pt too wide) detected at line 31413 [] [] [391] Overfull \hbox (7.54756pt too wide) detected at line 31462 [] [] [392] Overfull \hbox (22.23291pt too wide) detected at line 31532 [] [] [393] Underfull \hbox (badness 10000) in paragraph at lines 31581--31581 [] [394] Overfull \hbox (31.35318pt too wide) in paragraph at lines 31636--31639 []\OT1/cmr/bx/n/12 5) \OT1/cmr/m/n/12 Dies folgt aus 1.9., an-ge-wandt auf $\OM L/cmm/m/it/12 A \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 , $\OML/cm m/m/it/12 f \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^U$ \OT1/cmr/m/n/12 und $[][] = [][] \OML/cmm/m/it/12 :$ [] [395] Overfull \hbox (32.09752pt too wide) detected at line 31727 [] [] Overfull \hbox (3.1395pt too wide) in paragraph at lines 31727--31733 \OT1/cmr/m/n/12 Somit exis-tiert ge-nau ein Na-ka-ya-ma-au-to-mor-phis-mus $\OM L/cmm/m/it/12 ^^Z$ \OT1/cmr/m/n/12 der Fro-be-ni-u-sal-ge-bra $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 bez[]uglich [] Overfull \hbox (20.26865pt too wide) detected at line 31746 [] [] [396] [397] Overfull \hbox (20.17351pt too wide) detected at line 31876 \OML/cmm/m/it/12 p [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^U [] p [] \OT1/cmr/m /n/12 = [] [] = [] [] = \OML/cmm/m/it/12 p [] ^^U [] \OT1/cmr/m/n/12 = \OML/cmm /m/it/12 p [] : [] Overfull \hbox (40.86171pt too wide) detected at line 31935 [] [] Overfull \hbox (65.51611pt too wide) detected at line 31970 [] [] [398] Overfull \hbox (15.73514pt too wide) detected at line 31981 \OML/cmm/m/it/12 ^^Z [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 S [] ^^U [] \OT1/cmr /m/n/12 = \OML/cmm/m/it/12 S [] ^^U [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^U [ ] S [] ; [] Overfull \hbox (17.30461pt too wide) in paragraph at lines 32010--32015 \OT1/cmr/m/n/12 Doch $\OML/cmm/m/it/12 aS[] [] \OT1/cmr/m/n/12 = ^^C\OML/cmm/m/ it/12 ;$ \OT1/cmr/m/n/12 denn nach \OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 gilt f[] ur je-des $\OML/cmm/m/it/12 x$ \OT1/cmr/m/n/12 die Be-zie-hung $\OML/cmm/m/it/1 2 x \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^U [] aS[] []$\OT1/cmr/m/n/12 , [] [399] Overfull \hbox (31.55313pt too wide) detected at line 32093 [] \OML/cmm/m/it/12 ( q \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 ( q \OT1/cmr/m/n/ 12 = \OML/cmm/m/it/12 q [] x[]p [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 p * [] \O T1/cmr/m/n/12 = \OML/cmm/m/it/12 p * [] : [] Overfull \hbox (0.66893pt too wide) detected at line 32114 [] [] Overfull \hbox (3.59924pt too wide) detected at line 32167 [] [] [400] Overfull \hbox (53.38715pt too wide) detected at line 32209 [] [] Overfull \hbox (12.21527pt too wide) in paragraph at lines 32212--32215 []\OT1/cmr/bx/n/12 Definition: \OT1/cmr/m/n/12 Ei-ne end-lich-di-men-sio-na-le Hopfal-ge-bra $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 hei^^Yt \OT1/cmr/m/it/12 un- i-mo-du-lar\OT1/cmr/m/n/12 , wenn $[][] [] = [] [401] Overfull \hbox (30.75833pt too wide) in paragraph at lines 32308--32315 \OML/cmm/m/it/12 R[]$ \OT1/cmr/m/n/12 der freie $\OML/cmm/m/it/12 R$\OT1/cmr/m/ n/12 -Linksmodul mit Ba-sis $[][] \OML/cmm/m/it/12 :$ \OT1/cmr/m/n/12 De-fi-nie -re einen $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 -Linksmodulhomomorphismus [] [402] [403] Overfull \hbox (23.63513pt too wide) detected at line 32414 [] [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 xay \OT1/cmr/m/n/12 = \OML/cmm/m/it /12 uxayv \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 a \OT1/cmr/m/n/12 = [] \OML/cmm /m/it/12 a: [] [404] Overfull \hbox (50.35536pt too wide) detected at line 32549 [] \OML/cmm/m/it/12 f [] \OT1/cmr/m/n/12 = [] [] \OML/cmm/m/it/12 ax[] \OMS/cms y/m/n/12 \OML/cmm/m/it/12 y[] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 xax[] \OMS/cmsy/m/n /12 [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 xax[] \OMS/cmsy/m/n/12 [][] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 xayx[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 y[]; [] [405] [406] Overfull \hbox (39.83179pt too wide) in paragraph at lines 32671--32680 \OT1/cmr/m/n/12 von $\OML/cmm/m/it/12 H$ \OT1/cmr/m/it/12 als $\OML/cmm/m/it/12 H$\OT1/cmr/m/it/12 -Linksmodul \OT1/cmr/m/n/12 ist (denn $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ist hal-b-ein-fach), und da der $\OML/cmm/m/it/12 H$\OT1/cmr/m/ n/12 -Linksmodulepimorphismus [] [407] [408] Overfull \hbox (10.74756pt too wide) detected at line 32832 [] [] Overfull \hbox (57.20064pt too wide) detected at line 32846 [] [] [409] Overfull \hbox (6.18484pt too wide) detected at line 32882 [] [] [410] [411] Overfull \hbox (9.87317pt too wide) in paragraph at lines 33051--33055 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 F[]ur al-le $\OML/cmm/m/it/12 e \OMS/cmsy /m/n/12 2 \OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 \OMS/cmsy /m/n/12 2 \OML/cmm/m/it/12 k;$ \OT1/cmr/m/n/12 die $\OML/cmm/m/it/12 e[] \OT1/c mr/m/n/12 = \OML/cmm/m/it/12 e$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 S[] [] \ OT1/cmr/m/n/12 = \OML/cmm/m/it/12 e$ \OT1/cmr/m/n/12 erf[]ullen, gilt $\OML/cmm /m/it/12 [][] [] \OT1/cmr/m/n/12 = [] [412] Overfull \hbox (7.1922pt too wide) in paragraph at lines 33119--33126 \OT1/cmr/m/n/12 0$ und $\OML/cmm/m/it/12 ^^U [] \OMS/cmsy/m/n/12 6\OT1/cmr/m/n/ 12 = 0$ ist. Das hei^^Yt: Ge-nau dann sind $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 H[]$ \OT1/cmr/m/n/12 hal-b-ein-fach, wenn $\OML/cmm/m/it /12 ^^U [] " [] \OMS/cmsy/m/n/12 6\OT1/cmr/m/n/12 = [] [413] Overfull \hbox (69.04424pt too wide) detected at line 33265 [] [] [414] Overfull \hbox (58.38663pt too wide) in paragraph at lines 33306--33313 []\OT1/cmr/bx/n/12 Definition: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cm r/m/n/12 ei-ne Hopfal-ge-bra. Die Teil-men-ge $\OML/cmm/m/it/12 R [] \OT1/cmr/m /n/12 = []$ [] [415] [416] Overfull \hbox (20.83946pt too wide) in paragraph at lines 33465--33480 \OT1/cmr/m/n/12 Da $^^C$ ein Rechts-in-te-gral ist, ist aber $[]^^C\OML/cmm/m/i t/12 S [] \OT1/cmr/m/n/12 = []\OML/cmm/m/it/12 " [] \OT1/cmr/m/n/12 ^^C = [] ^^ C = [] [417] Overfull \hbox (26.97438pt too wide) in paragraph at lines 33519--33532 \OT1/cmr/m/n/12 Da $^^@$ ein Links-in-te-gral ist, ist aber $^^C[]^^@\OML/cmm/m /it/12 S [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 " [] \OT1/cmr/m/n/12 ^^@\OML/cmm /m/it/12 S [] \OT1/cmr/m/n/12 = ^^@\OML/cmm/m/it/12 S [] \OT1/cmr/m/n/12 = [] Overfull \hbox (8.56328pt too wide) in paragraph at lines 33519--33532 \OT1/cmr/m/n/12 ^^@\OML/cmm/m/it/12 S [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 " [ ] S []$ \OT1/cmr/m/n/12 (denn $\OML/cmm/m/it/12 S []$ \OT1/cmr/m/n/12 ist ein L inks-in-te-gral), al-so $\OML/cmm/m/it/12 ^^_[] [] \OT1/cmr/m/n/12 = \OML/cmm/m /it/12 ^^U [] \OT1/cmr/m/n/12 = [] Overfull \hbox (18.55888pt too wide) detected at line 33584 [] [] [418] Overfull \hbox (4.71104pt too wide) in paragraph at lines 33617--33619 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 Da wir ihn jetzt mehr-mals ver-wen-det ha -ben, soll-ten wir den Satz von Wedderburn- [] Overfull \hbox (7.72925pt too wide) in paragraph at lines 33638--33640 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne end-lich-di-men-sio-na-le Hopfal-ge-bra, und sei $\OML/cmm/m/it/12 k \OMS /cmsy/m/n/12 ^^R \OML/cmm/m/it/12 `$ \OT1/cmr/m/n/12 ei-ne K[]orpererweiterung. [] [419] Overfull \hbox (16.6248pt too wide) in paragraph at lines 33714--33721 \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 ^^W[] \OMS/cmsy/m/n/12 2 []$ \OT1/cmr/m/n /12 f[]ur al-le $1 \OMS/cmsy/m/n/12 ^^T \OML/cmm/m/it/12 j \OMS/cmsy/m/n/12 ^^T \OML/cmm/m/it/12 m$ \OT1/cmr/m/n/12 (denn $[][] = []$). Folg-lich ist $[] [] \ OMS/cmsy/m/n/12 2 [] [420] Overfull \hbox (33.09631pt too wide) detected at line 33773 [] [] Overfull \hbox (7.88995pt too wide) in paragraph at lines 33776--33795 []\OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 i \OMS/cmsy/m/n/12 2 []$ \OT1/cmr/m/n/1 2 be-lie-big. Nach Wedderburn-Artin ist $\OML/cmm/m/it/12 C[] [] [][] [] [] [][ ] []$ [] [421] Overfull \hbox (29.65355pt too wide) detected at line 33898 [] [] [422] [423] Overfull \hbox (6.73842pt too wide) detected at line 34033 [] [] Overfull \hbox (50.8852pt too wide) detected at line 34069 [] \OML/cmm/m/it/12 f \OT1/cmr/m/n/12 = [] []\OML/cmm/m/it/12 f \OMS/ cmsy/m/n/12 2 [] \OML/cmm/m/it/12 V [] Overfull \hbox (70.63663pt too wide) detected at line 34114 [] [] [424] Overfull \hbox (19.95482pt too wide) detected at line 34150 [] [] Overfull \hbox (68.01633pt too wide) detected at line 34165 [] [] [425] [426] Overfull \hbox (12.22194pt too wide) detected at line 34336 [] [] [427] Overfull \hbox (32.11566pt too wide) in paragraph at lines 34416--34430 \OML/cmm/m/it/12 k []$ \OT1/cmr/m/n/12 von $\OML/cmm/m/it/12 k []$\OT1/cmr/m/n/ 12 -Linksmoduln. We-gen $[] [] = [] \OML/cmm/m/it/12 k [] [] \OT1/cmr/m/n/12 = [] [428] [429] Overfull \hbox (18.9088pt too wide) detected at line 34570 [] [] [430] Overfull \hbox (47.86984pt too wide) detected at line 34702 [] [] [431] [432] Overfull \hbox (6.22523pt too wide) in paragraph at lines 34790--34795 []\OT1/cmr/bx/n/12 e) \OT1/cmr/m/n/12 Es gibt ein $\OML/cmm/m/it/12 t \OMS/cmsy /m/n/12 ^^U \OT1/cmr/m/n/12 1$ und Schiefk[]orper $\OML/cmm/m/it/12 D[];$ $D[]; $ $:::;$ $D[]$ \OT1/cmr/m/n/12 so-wie gan-ze Zah-len $\OML/cmm/m/it/12 n[]; n[] ; :::; n[] \OMS/cmsy/m/n/12 ^^U [] Overfull \hbox (0.27365pt too wide) in paragraph at lines 34807--34818 \OT1/cmr/m/n/12 Schiefk[]orper $\OML/cmm/m/it/12 D[];$ $D[];$ $:::;$ $D[]$ \OT1 /cmr/m/n/12 so-wie gan-ze Zah-len $\OML/cmm/m/it/12 n[]; n[]; :::; n[] \OMS/cms y/m/n/12 ^^U \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ;$ \OT1/cmr/m/n/12 f[]ur die gil t: $\OML/cmm/m/it/12 A\U/msb/m/n/12 ^^^ [] \OML/cmm/m/it/12 A [] [] [433] Overfull \hbox (44.09807pt too wide) in paragraph at lines 34884--34890 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/bx/n/12 1) $\OT1/cmr/m/n/12 =[]\OMS/cmsy/m/ n/12 )\OT1/cmr/m/n/12 :$ Nach Vor-aus-set-zung ist $\OML/cmm/m/it/12 n \OT1/cmr /m/n/12 = \OML/cmm/m/it/12 r[] [] \OT1/cmr/m/n/12 = []\OML/cmm/m/it/12 ;$ \OT1/ cmr/m/n/12 al-so $\OML/cmm/m/it/12 `[] [] \OT1/cmr/m/n/12 = [] Overfull \hbox (20.16951pt too wide) in paragraph at lines 34902--34914 []\OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 F[]ur al-le $\OML/cmm/m/it/12 Q \OMS/cmsy /m/n/12 2 [] \OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 n \OT1/c mr/m/n/12 = \OML/cmm/m/it/12 r[] [] \OMS/cmsy/m/n/12 ^^U \OML/cmm/m/it/12 r[] [ ] \OT1/cmr/m/n/12 = []\OML/cmm/m/it/12 ;$ \OT1/cmr/m/n/12 al-so $\OML/cmm/m/it/ 12 n`[] [] \OMS/cmsy/m/n/12 ^^U [] Overfull \hbox (44.03828pt too wide) in paragraph at lines 34921--34926 \OT1/cmr/m/n/12 wobei der Iso-mor-phis-mus $\OML/cmm/m/it/12 V\U/msb/m/n/12 ^^^ \OML/cmm/m/it/12 V [] A \OMS/cmsy/m/n/12 ! [] \OML/cmm/m/it/12 V\U/msb/m/n/12 ^ ^^\OML/cmm/m/it/12 VQ$ \OT1/cmr/m/n/12 de-fi-niert ist durch $[] \OMS/cmsy/m/n/ 12 7! [][] \OML/cmm/m/it/12 ;$ [] Overfull \hbox (17.76431pt too wide) in paragraph at lines 34930--34938 \OML/cmm/m/it/12 V\U/msb/m/n/12 ^^^\OML/cmm/m/it/12 V [] A$ \OT1/cmr/m/n/12 dur ch die-ses Dia-gramm de-fi-niert; die-ses $\OML/cmm/m/it/12 f$ \OT1/cmr/m/n/12 ist ein $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodulepimorphismus [] [434] [435] Overfull \hbox (59.0579pt too wide) detected at line 35051 [] [] Overfull \hbox (6.69624pt too wide) in paragraph at lines 35061--35066 \OT1/cmr/bx/n/12 Definition: 1) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/c mr/m/n/12 ei-ne Hopfal-ge-bra, und sei $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei- ne $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra verm[]oge [] Overfull \hbox (0.59412pt too wide) in paragraph at lines 35086--35091 []\OT1/cmr/bx/n/12 4.7. Lem-ma: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/c mr/m/n/12 ei-ne Hopfal-ge-bra, und sei $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei- ne $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulunteralgebra [] [436] Overfull \hbox (0.26448pt too wide) in paragraph at lines 35124--35131 []\OT1/cmr/m/it/12 Bedingung $\OMS/cmsy/m/n/12 C[]$\OT1/cmr/m/it/12 : \OT1/cmr/ m/n/12 Die Ab-bil-dung $\OML/cmm/m/it/12 ^^Z$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/i t/12 A$\OT1/cmr/m/n/12 -rechtslinear, wo-bei die $\OML/cmm/m/it/12 A$\OT1/cmr/m /n/12 -Rechtsmodulstruktur [] Overfull \hbox (14.23045pt too wide) in paragraph at lines 35124--35131 \OT1/cmr/m/n/12 auf-ge-fasst wird (weil $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ei n $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodul ist), und aus die-ser $\OML/ cmm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmodulstruktur [] Overfull \hbox (5.42894pt too wide) in paragraph at lines 35143--35146 \OT1/cmr/m/n/12 einen Vek-tor-raum-ho-mo-mor-phis-mus, der so-wohl $\OML/cmm/m/ it/12 A$\OT1/cmr/m/n/12 -rechtslinear, als auch $\OML/cmm/m/it/12 H$\OT1/cmr/m/ n/12 -rechtscolinear [] Overfull \hbox (5.207pt too wide) in paragraph at lines 35166--35171 \OT1/cmr/m/n/12 Modulerzeugendensystem von $\OML/cmm/m/it/12 V:$ \OT1/cmr/m/n/1 2 Dann ist $[][]$ ein $\OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Modulerzeugendensystem [] [437] Overfull \hbox (57.00368pt too wide) in paragraph at lines 35196--35206 []\OT1/cmr/bx/n/12 4) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne Hopfal-ge-bra, und sei $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei-ne Rechts- co-mo-du-lal-ge-bra mit $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulstruk tur [] Overfull \hbox (3.35909pt too wide) in paragraph at lines 35241--35252 []\OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei-ne end-lich-di-me n-sio-na-le $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra[][][]. Se i $\OML/cmm/m/it/12 P \U/msa/m/n/12 C \OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 mit $ \OML/cmm/m/it/12 I[] [] \OT1/cmr/m/n/12 = [] Overfull \hbox (5.41406pt too wide) in paragraph at lines 35241--35252 \OT1/cmr/m/n/12 Rechtsmodul er-zeu-gen. An-ge-nom-men, $[]\OML/cmm/m/it/12 ; [] ; :::; []$ \OT1/cmr/m/n/12 sei ei-ne Ba-sis des $\OML/cmm/m/it/12 A\U/msb/m/n/1 2 ^^^\OML/cmm/m/it/12 P$\OT1/cmr/m/n/12 -Rechtsmoduls [] [438] Overfull \hbox (15.46346pt too wide) in paragraph at lines 35308--35315 []\OT1/cmr/m/n/12 Nach 4.8. \OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 ist $\OML/cmm/m /it/12 ^^Z [] ; ^^Z [] ; :::; ^^Z []$ \OT1/cmr/m/n/12 ein Er-zeu-gen-den-sys-te m des $\OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmoduls [] Overfull \hbox (25.9419pt too wide) in paragraph at lines 35308--35315 \OML/cmm/m/it/12 V \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H:$ \OT1/cmr/m/n/12 So-mit ist $[]\OML/cmm/m/it/12 ; []; :::; []$ \OT1/cmr/m/n/12 ein Er-zeu-gen-den-sys-tem des $[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmoduls [] Overfull \hbox (2.6482pt too wide) in paragraph at lines 35316--35326 []\OT1/cmr/m/n/12 Aber nach Vor-aus-set-zung ist $[]\OML/cmm/m/it/12 ; []; :::; []$ \OT1/cmr/m/n/12 ei-ne Ba-sis des $\OML/cmm/m/it/12 A\U/msb/m/n/12 ^^^\OML/ cmm/m/it/12 P$\OT1/cmr/m/n/12 -Rechtsmoduls $\OML/cmm/m/it/12 V\U/msb/m/n/12 ^^ ^\OML/cmm/m/it/12 VP$\OT1/cmr/m/n/12 . [] Overfull \hbox (8.13985pt too wide) in paragraph at lines 35316--35326 \OT1/cmr/m/n/12 Mit an-de-ren Wor-ten: $[]\OML/cmm/m/it/12 ; []; :::; []$ \OT1/ cmr/m/n/12 ist ei-ne Ba-sis des $[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtsmoduls [] Overfull \hbox (34.30617pt too wide) in paragraph at lines 35327--35343 \OT1/cmr/m/n/12 ein Er-zeu-gen-den-sys-tem und hat ge-nau-so vie-le Ele-men-te wie die Ba-sis $[]\OML/cmm/m/it/12 ; []; :::; []$\OT1/cmr/m/n/12 ), [] [439] Overfull \hbox (0.1874pt too wide) detected at line 35375 [] [] Overfull \hbox (16.99997pt too wide) in paragraph at lines 35375--35380 \OT1/cmr/m/n/12 f[]ur al-le $\OML/cmm/m/it/12 Q \OMS/cmsy/m/n/12 2 [] \OML/cmm/ m/it/12 A:$ \OT1/cmr/m/n/12 Nach Lem-ma 4.5. \OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/1 2 gibt es al-so ein $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodulerzeugenden system [] [440] Overfull \hbox (30.77869pt too wide) in paragraph at lines 35493--35497 []\OT1/cmr/m/n/12 Als \OT1/cmr/m/it/12 Ob-jekt von $[]\OMS/cmsy/m/n/12 M[]$ \OT 1/cmr/m/n/12 be-zeich-nen wir je-den Vek-tor-raum $\OML/cmm/m/it/12 V$ \OT1/cmr /m/n/12 mit ei-ner $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmodulstruktur [] Overfull \hbox (1.8278pt too wide) in paragraph at lines 35498--35505 \OT1/cmr/m/n/12 auf-ge-fasst wird (weil $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ei n $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Linksmodul ist), und aus die-ser $\OML/c mm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulstruktur [] [441] Overfull \hbox (31.50926pt too wide) detected at line 35587 [] [] Overfull \hbox (14.58464pt too wide) detected at line 35602 [] [] Overfull \hbox (38.5261pt too wide) detected at line 35625 [] [] [442] Overfull \hbox (20.33278pt too wide) in paragraph at lines 35682--35686 []\OT1/cmr/bx/n/12 4.14. Lem-ma: \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 A$ \OT1/ cmr/m/n/12 ei-ne end-lich-di-men-sio-na-le $\OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -Algebra, und sei $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 ein $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 -Rechtsmodul. [] Overfull \hbox (17.74004pt too wide) in paragraph at lines 35720--35725 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/m/n/12 Nach 4.7. und 4.12. (\OT1/cmr/bx/n/1 2 3) $\OT1/cmr/m/n/12 =[]\OMS/cmsy/m/n/12 )$ \OT1/cmr/bx/n/12 1)\OT1/cmr/m/n/12 ) ist $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei-ne $\OML/cmm/m/it/12 H$\OT1/cmr/ m/n/12 -einfache $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra. [] [443] [444] Overfull \hbox (7.34575pt too wide) in paragraph at lines 35812--35815 \OT1/cmr/m/n/12 ein Iso-mor-phis-mus von Vek-torr[]aumen (und so-gar ein Iso-mo r-phis-mus von $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmoduln [] [445] [446] Overfull \hbox (14.06538pt too wide) detected at line 35977 [] [] [447] [448] Overfull \hbox (30.62135pt too wide) detected at line 36082 [] [] Overfull \hbox (30.62135pt too wide) detected at line 36089 [] [] [449] Overfull \hbox (25.1222pt too wide) in paragraph at lines 36222--36229 \OT1/cmr/m/n/12 exakt (wo-bei $\OML/cmm/m/it/12 K[]$ \OT1/cmr/m/n/12 als Fak-to r-vek-tor-raum von $\OML/cmm/m/it/12 H[]$ \OT1/cmr/m/n/12 auf-ge-fasst wird). H ieraus folgt $[][] = [] [450] Overfull \hbox (16.76247pt too wide) in paragraph at lines 36233--36240 \OT1/cmr/m/n/12 torr[]aume, auf de-nen so-wohl $\OML/cmm/m/it/12 A$\OT1/cmr/m/n /12 -Linksmodulstrukturen als auch $\OML/cmm/m/it/12 C$\OT1/cmr/m/n/12 -Rechtsc omodulstrukturen [] Overfull \hbox (5.24199pt too wide) detected at line 36311 [] [] [451] [452] Overfull \hbox (12.61118pt too wide) in paragraph at lines 36482--36556 []\OT1/cmr/m/n/12 Die Ab-bil-dung $^^H \OMS/cmsy/m/n/12 ^^N [][] \OT1/cmr/m/n/1 2 : \OML/cmm/m/it/12 H \OMS/cmsy/m/n/12 [] \OML/cmm/m/it/12 H \OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 H \OMS/cmsy/m/n/12 []$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 K$\OT1/cmr/m/n/12 -linkslinear[][][] und $[]$-rechtscolinear[][][]. [] [453] Overfull \hbox (44.46817pt too wide) in paragraph at lines 36583--36606 \OT1/cmr/m/n/12 ^^D$ ein Iso-mor-phis-mus von $\OML/cmm/m/it/12 H[] \OMS/cmsy/m /n/12 ! [][]$\OT1/cmr/m/n/12 , wel-cher gleich-zei-tig ein $\OML/cmm/m/it/12 K$ \OT1/cmr/m/n/12 -Linksmodulisomorphismus [] Overfull \hbox (28.22617pt too wide) in paragraph at lines 36583--36606 \OT1/cmr/m/n/12 und ein $[]$-Rechtscomodulisomorphismus ist (denn er ist $\OML/ cmm/m/it/12 K$\OT1/cmr/m/n/12 -linkslinear und $[]$-rechtscolinear). [] [454] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 36659. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 36659. [455] [456] Overfull \hbox (26.86331pt too wide) in paragraph at lines 36767--36775 []\OT1/cmr/m/n/12 F[]ur je-des $\OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 2 []$ \OT1/ cmr/m/n/12 sei $\OML/cmm/m/it/12 ^^Y[] \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 W[] \ OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 die Sur-jek-ti-on, die du rch $\OML/cmm/m/it/12 ^^Y[] [] \OT1/cmr/m/n/12 = [] Overfull \hbox (5.4233pt too wide) in paragraph at lines 36776--36781 []\OT1/cmr/m/n/12 Somit ist $\OML/cmm/m/it/12 ^^Y[] \OMS/cmsy/m/n/12 ^^N \OML/c mm/m/it/12 f \OMS/cmsy/m/n/12 ^^N \OML/cmm/m/it/12 ^^^[] \OT1/cmr/m/n/12 : \OML /cmm/m/it/12 V \OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 ein von $ 0$ ver-schie-de-ner $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulhomomorphism us. [] [457] Overfull \hbox (9.43681pt too wide) in paragraph at lines 36938--36940 []\OT1/cmr/m/it/12 Beweis von Lem-ma 2: \OT1/cmr/m/n/12 Wir kon-stru-ie-ren die Ab-bil-dung $\OML/cmm/m/it/12 ^^Y \OT1/cmr/m/n/12 : [] \OMS/cmsy/m/n/12 ! []$ [] [458] [459] [460] Overfull \hbox (56.3327pt too wide) in paragraph at lines 37152--37157 \OT1/cmr/m/n/12 zueinander in-ver-se Vek-tor-rau-mi-so-mor-phis-men sind. Da-he r ist $[][] [] [] [][] [] \OML/cmm/m/it/12 ;$ [] Overfull \hbox (37.89824pt too wide) detected at line 37166 [] [] Overfull \hbox (56.98547pt too wide) in paragraph at lines 37213--37218 \OT1/cmr/m/n/12 zueinander in-ver-se Vek-tor-rau-mi-so-mor-phis-men sind. Da-he r ist $[][] [] [] [][] [] \OML/cmm/m/it/12 ;$ [] Overfull \hbox (38.55101pt too wide) detected at line 37227 [] [] [461] Overfull \hbox (7.00455pt too wide) in paragraph at lines 37245--37245 []\OT1/cmr/bx/n/17.28 IV. Ka-pi-tel: Drinfeld-Doppel und Bo-so-ni-sie-rung [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 37247. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 37247. [462] Overfull \hbox (64.19858pt too wide) detected at line 37381 [] [] [463] Overfull \hbox (8.90512pt too wide) detected at line 37417 [] [] \OT1/cmr/m/n/12 = [] = [] = [] [] Overfull \hbox (51.28806pt too wide) in paragraph at lines 37471--37474 []\OT1/cmr/bx/n/12 4) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne Hopfal-ge-bra. Sei $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ei-ne $\OML/cmm/m /it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra mit $\OML/cmm/m/it/12 H$\OT1/cmr /m/n/12 -Rechtscomodulstruktur [] Overfull \hbox (14.44647pt too wide) in paragraph at lines 37475--37478 []\OT1/cmr/m/n/12 Wir neh-men fer-ner an, $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 sei $\OML/cmm/m/it/12 H$\OT1/cmr/m/it/12 -cleft\OT1/cmr/m/n/12 , d. h. es ge-be ei-ne $\OMS/cmsy/m/n/12 ^^C$\OT1/cmr/m/n/12 -invertierbare $\OML/cmm/m/it/12 H $\OT1/cmr/m/n/12 -rechtscolineare [] [464] [465] Overfull \hbox (4.76553pt too wide) detected at line 37625 [] [] Overfull \hbox (0.57457pt too wide) detected at line 37648 [] [] [466] Overfull \hbox (66.4528pt too wide) in paragraph at lines 37718--37724 []\OT1/cmr/m/it/12 Warnung: \OT1/cmr/m/n/12 Auch wenn wir ge-ra-de das Zei-chen $\OML/cmm/m/it/12 ]$ \OT1/cmr/m/n/12 ver-wen-den, soll-te un-se-re $\OML/cmm/m /it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra [] Overfull \hbox (33.70116pt too wide) in paragraph at lines 37725--37727 []\OT1/cmr/m/n/12 Wir m[]ussen be-wei-sen, da^^Y das so de-fi-nier-te $\OML/cmm /m/it/12 k][]H$ \OT1/cmr/m/n/12 tats[]achlich ei-ne $\OML/cmm/m/it/12 H$\OT1/cm r/m/n/12 -Rechtscomodulalgebra [] Overfull \hbox (20.42653pt too wide) detected at line 37742 [] [] Overfull \hbox (20.71817pt too wide) detected at line 37752 [] [] \OT1/cmr/m/n/12 = [] [] = \OML/cmm/m/it/12 ^^[ [] ^^[ [] ]x[]y[]z[]: [] [467] Overfull \hbox (1.91275pt too wide) in paragraph at lines 37787--37789 []\OT1/cmr/bx/n/12 b) \OT1/cmr/m/n/12 Au-^^Yer-dem ist $\OML/cmm/m/it/12 k][]H$ \OT1/cmr/m/n/12 ei-ne $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -clefte $\OML/cmm/m/ it/12 H$\OT1/cmr/m/n/12 -Rechtscomodulalgebra; die $\OML/cmm/m/it/12 H$\OT1/cmr /m/n/12 -rechtscolineare [] Overfull \hbox (4.56279pt too wide) detected at line 37814 [] [] Overfull \hbox (56.33769pt too wide) detected at line 37894 [] [] [468] Overfull \hbox (15.65834pt too wide) in paragraph at lines 37908--37912 \OT1/cmr/m/n/12 ist of-fen-sicht-lich auch ein $\OML/cmm/m/it/12 H$\OT1/cmr/m/n /12 -Rechtscomodulhomomorphismus (denn $\OML/cmm/m/it/12 j$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -rechtscolinear) [] Overfull \hbox (55.75906pt too wide) detected at line 37971 [] [] [469] Overfull \hbox (43.85881pt too wide) detected at line 38077 [] [] [470] Overfull \hbox (43.85881pt too wide) detected at line 38189 [] [] [471] Overfull \hbox (27.95935pt too wide) in paragraph at lines 38209--38214 []\OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 F[]ur al-le $\OML/cmm/m/it/12 x; y; z \OM S/cmsy/m/n/12 2 \OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/12 ^^[[] [] ^^[[] [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 ^^[[] [] ^^[[] [] :$ [] Overfull \hbox (14.00894pt too wide) in paragraph at lines 38219--38223 []\OT1/cmr/bx/n/12 5) \OT1/cmr/m/n/12 Falls die Bi-al-ge-bra $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne An-ti-po-de $\OML/cmm/m/it/12 S$ \OT1/cmr/m/n/12 hat, ist $\OML/cmm/m/it/12 ^^[ [] ^^[[] [] \OT1/cmr/m/n/12 = [] [472] Overfull \hbox (32.59297pt too wide) detected at line 38245 [] [] Overfull \hbox (43.51225pt too wide) detected at line 38282 [] [] Overfull \hbox (5.98201pt too wide) detected at line 38329 [] [] Overfull \hbox (20.13017pt too wide) detected at line 38359 [] [] [473] Overfull \hbox (35.6852pt too wide) detected at line 38518 [] [] [474] [475] Overfull \hbox (46.41284pt too wide) detected at line 38615 [] [] Overfull \hbox (2.25908pt too wide) detected at line 38691 [] [] [476] [477] Overfull \hbox (19.33257pt too wide) in paragraph at lines 38774--38780 \OT1/cmr/m/n/12 und ana-log $\OML/cmm/m/it/12 ^^\ [] ^^\ [] \OT1/cmr/m/n/12 = \ OML/cmm/m/it/12 " [] " [] ;$ \OT1/cmr/m/n/12 und da-her ist $\OML/cmm/m/it/12 u \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 a \OMS/cmsy/m/n/12 7! \OML/cmm/m/it/12 ^^\ []$ [] [478] Overfull \hbox (27.24728pt too wide) in paragraph at lines 38906--38916 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/bx/n/12 1) \OT1/cmr/m/n/12 Sei-en $\OML/cmm /m/it/12 x; y; z \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 U \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 A:$ \OT1/cmr/m/n/12 Wir m[]ussen nach-wei-sen, da^^Y $\OML/cm m/m/it/12 ^^[ [] ^^[ [] \OT1/cmr/m/n/12 = [] Overfull \hbox (33.14418pt too wide) detected at line 38941 [] [] Overfull \hbox (31.39072pt too wide) detected at line 38970 [] [] [479] [480] Overfull \hbox (43.02403pt too wide) detected at line 39096 [] [] [481] [482] Overfull \hbox (13.29968pt too wide) detected at line 39248 \OML/cmm/m/it/12 e[]u \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 e[]e[]g \OT1/cmr/m/ n/12 = \OML/cmm/m/it/12 e[]h[] [] ue[] \OT1/cmr/m/n/12 = [] \ OML/cmm/m/it/12 e[] [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 e[]e[]g \OT1/cmr/m /n/12 = \OML/cmm/m/it/12 e[]h[] [] [483] Overfull \hbox (32.18344pt too wide) detected at line 39353 [] [] [484] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 39433. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `superscript' on input line 39433. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 39433. [485] Overfull \hbox (63.56276pt too wide) detected at line 39511 [] [] Overfull \hbox (57.51115pt too wide) in paragraph at lines 39544--39551 \OT1/cmr/m/n/12 nach dem Ho-mo-mor-phie-satz, und so-mit ist $[] [] = [] [] \OM S/cmsy/m/n/12 ^^T [] [486] Overfull \hbox (50.48401pt too wide) in paragraph at lines 39557--39560 []\OT1/cmr/m/n/12 Nach \OT1/cmr/bx/n/12 4) \OT1/cmr/m/n/12 ist $\OML/cmm/m/it/1 2 A\U/msb/m/n/12 ^^^\OML/cmm/m/it/12 I$ \OT1/cmr/m/n/12 ein end-lich-di-men-sio -na-ler $\OML/cmm/m/it/12 A$\OT1/cmr/m/n/12 -Rechtsmodul. So-mit ist auch $[] [ ] \OML/cmm/m/it/12 < [] Overfull \hbox (24.65346pt too wide) detected at line 39583 [] [] \OT1/cmr/m/n/12 = [] [] \OMS/cmsy/m/n/12 ^^T [] [] \OML/cmm/m/it/12 < \OM S/cmsy/m/n/12 1 [] Overfull \hbox (26.38802pt too wide) in paragraph at lines 39610--39623 []\OT1/cmr/m/n/12 Nun ist $\OML/cmm/m/it/12 f$ \OT1/cmr/m/n/12 ein Ele-ment von $[][]$ (denn $\OML/cmm/m/it/12 f [] \OT1/cmr/m/n/12 = 0$ und $[][] = []$). [] Overfull \hbox (7.5449pt too wide) detected at line 39635 [] [] [487] Overfull \hbox (7.55637pt too wide) detected at line 39723 [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 : [] [488] Overfull \hbox (8.99924pt too wide) detected at line 39804 [] [] [489] Overfull \hbox (16.06479pt too wide) detected at line 39926 [] [] [490] Overfull \hbox (1.79547pt too wide) in paragraph at lines 39961--39966 \OT1/cmr/m/n/12 Daher ist $\OML/cmm/m/it/12 c[] \OT1/cmr/m/n/12 + \OML/cmm/m/it /12 c[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 c[]$\OT1/cmr/m/n/12 . Nach den De-fi -ni-tio-nen von $\OML/cmm/m/it/12 C []$\OT1/cmr/m/n/12 , $\OML/cmm/m/it/12 C [] $ \OT1/cmr/m/n/12 und $\OML/cmm/m/it/12 C []$ [] Overfull \hbox (4.03583pt too wide) detected at line 40017 \OML/cmm/m/it/12 c[] [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 f [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 f [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 f [] \OT1/cmr/m/n/1 2 = [] [] = \OML/cmm/m/it/12 g [] [] [491] Overfull \hbox (12.87636pt too wide) in paragraph at lines 40029--40032 []\OT1/cmr/bx/n/12 6) \OT1/cmr/m/n/12 Sei $\OML/cmm/m/it/12 H$ \OT1/cmr/m/n/12 ei-ne Bi-al-ge-bra, und sei-en $\OML/cmm/m/it/12 V$ \OT1/cmr/m/n/12 und $\OML/c mm/m/it/12 W$ \OT1/cmr/m/n/12 zwei end-lich-di-men-sio-na-le $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmoduln. [] Overfull \hbox (10.51352pt too wide) in paragraph at lines 40049--40055 \OT1/cmr/m/n/12 folgt hieraus schnell $\OML/cmm/m/it/12 C [] \OMS/cmsy/m/n/12 ^ ^A \OML/cmm/m/it/12 C [] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 C []$\OT1/cmr/m/n/1 2 , denn nach der De-fi-ni-ti-on von $\OML/cmm/m/it/12 C []$ [] Overfull \hbox (54.60204pt too wide) detected at line 40075 [] [] [492] Overfull \hbox (1.94147pt too wide) in paragraph at lines 40193--40196 []\OT1/cmr/m/it/12 Bemerkung: \OT1/cmr/m/n/12 Nat[]urlich las-sen sich die Punk -te \OT1/cmr/bx/n/12 2) \OT1/cmr/m/n/12 und \OT1/cmr/bx/n/12 3) \OT1/cmr/m/n/12 von 3.5$[]$ auch oh-ne R[]uckgriff [] [493] Overfull \hbox (18.84752pt too wide) in paragraph at lines 40241--40241 []\OT1/cmr/bx/n/14.4 Abschnitt 4. Drinfeld-Doppel und qua-si-tri-an-gul[]are Ho pfal- [] [494] Overfull \hbox (37.43184pt too wide) in paragraph at lines 40342--40349 []\OT1/cmr/m/n/12 Wende $[] []$ auf (Q3) an[][][], und er-hal-te $\OML/cmm/m/it /12 R[] \OMS/cmsy/m/n/12 [][] \OML/cmm/m/it/12 S[] [] \OT1/cmr/m/n/12 = [] [495] [496] [497] Overfull \hbox (51.59174pt too wide) detected at line 40523 [] [] Overfull \hbox (54.06393pt too wide) detected at line 40557 [] [] [498] [499] [500] Overfull \hbox (11.18365pt too wide) in paragraph at lines 40802--40806 \OML/cmm/m/it/12 k$\OT1/cmr/m/n/12 -lineare Ab-bil-dung von $\OML/cmm/m/it/12 V $ \OT1/cmr/m/n/12 nach $\OML/cmm/m/it/12 W$\OT1/cmr/m/n/12 , so hei^^Yt $\OML/c mm/m/it/12 f$ \OT1/cmr/m/n/12 ein \OT1/cmr/m/it/12 Yetter-Drinfeld-Modulhomomor phismus [] Overfull \hbox (62.33337pt too wide) in paragraph at lines 40802--40806 \OT1/cmr/m/n/12 ge-nau dann, wenn $\OML/cmm/m/it/12 f$ \OT1/cmr/m/n/12 ein $\OM L/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulhomomorphismus und ein $\OML/cmm/m/i t/12 H$\OT1/cmr/m/n/12 -Linkscomodulhomomorphismus [] [501] Overfull \hbox (33.8639pt too wide) in paragraph at lines 40811--40817 \OT1/cmr/m/n/12 Linkscomodulstruktur auf $\OML/cmm/m/it/12 V \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 W$ \OT1/cmr/m/n/12 die Dia-go-nal-struk-tu-ren sind, d. h. di e $\OML/cmm/m/it/12 H$\OT1/cmr/m/n/12 -Linksmodulstruktur [] Overfull \hbox (28.04778pt too wide) in paragraph at lines 40866--40874 \OML/cmm/m/it/12 ^^S$\OT1/cmr/m/n/12 ). Sei $\OML/cmm/m/it/12 R \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 A[]$\OT1/cmr/m/n/12 , wo-bei $\OML/cmm/m/it/12 A[]$ \OT1/cm r/m/n/12 als die Teil-men-ge $[]$ [] [502] Overfull \hbox (51.8601pt too wide) detected at line 40901 [] [] Overfull \hbox (20.7932pt too wide) in paragraph at lines 40922--40925 []\OT1/cmr/m/it/12 Beweis: \OT1/cmr/bx/n/12 1) \OT1/cmr/m/n/12 Die Ab-bil-dung $\OML/cmm/m/it/12 #$ \OT1/cmr/m/n/12 ist wohl-de-fi-niert, d. h. f[]ur je-des $ \OML/cmm/m/it/12 a \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ist $ \OML/cmm/m/it/12 a[] [] [] \OMS/cmsy/m/n/12 2 [] [503] [504] (./hopf.aux) Package rerunfilecheck Info: File `hopf.out' has not changed. (rerunfilecheck) Checksum: 471840D2ACB35A6CB95EC1E5B4862BEB;23345. ) Here is how much of TeX's memory you used: 14528 strings out of 478287 212583 string characters out of 5849289 611985 words of memory out of 5000000 31435 multiletter control sequences out of 15000+600000 486721 words of font info for 98 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 60i,29n,63p,3230b,807s stack positions out of 5000i,500n,10000p,200000b,80000s {/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-ts1.enc} Output written on hopf.pdf (504 pages, 2629282 bytes). PDF statistics: 3888 PDF objects out of 4296 (max. 8388607) 3260 compressed objects within 33 object streams 1046 named destinations out of 1200 (max. 500000) 785 words of extra memory for PDF output out of 10000 (max. 10000000)