A list of errata in the notes "Introduction to Abstract Algebra" by Samir Siksek ( https://homepages.warwick.ac.uk/staff/S.Siksek/teaching/aa/aanotes.pdf ) in the version of 2015-11-02:
[!!! UPDATE !!!
This list is no longer up-to-date. To my knowledge, they have all been corrected on 2019-10-18.]
- page 5, the displayed equation at the end: Is it a British standard
to write decimal points as $\cdots$ instead of $.$s? I'd find it a bit
confusing if I didn't already know what you mean.
- page 10, Example III.7: This is not obvious (see
https://proofwiki.org/wiki/General_Associativity_Theorem ). While the
structure of your notes makes it hard to include a proof at this
point, a reference to a proof or a proof in an appendix would be
highly helpful.
- page 24, footnote 1: "if an only if" -> "if and only if".
- page 32: "effect $\rho_0$" -> "effect as $\rho_0$".
- page 38, §VI.I: Again you are using general associativity (Example
III.7) here.
- page 40, §VI.2: I'd point out that part of additive notation is
writing $a/b$ as $a-b$.
- page 43, footnote 1: "where $0 \leq 0 < m$" -> "where $0 \leq a < m$".
- page 43, footnote 1: "the observe" -> "then observe".
- page 67: In the middle of the page, "\{..., -2k, -k, 0, k, 2k\}$" is
missing an "..." at the end.
- page 76, Example XII.13: "lets" -> "let's".
- page 82, Example XIII.2: I'd put a question mark after "What happens
to $f(\theta)$ as $\theta$ changes", unless you intend to convey that
it is a boring question.
- page 86, Exercise XIII.12: "of orders $3$" -> "of order $3$".
- page 87, proof of Lemma XIII.16: "as it a sum" -> "as it is a sum".
- page 96, §XIV.5: No need to require $n \geq 2$ in this discussion.
- page 98: "product of two permutation" -> "product of two permutations".
- page 98: "and example" -> "an example".
- page 98, Example XIV.24: "are repeatedly apply" -> "and repeatedly apply".
- page 98, Example XIV.24: "as either of" -> "in either of".
- page 102, §XIV.8: "the define" -> "then define".
- page 105, Example XIV.39: "of $4$ transposition" -> "of $4$ transpositions".
- page 108: "number moves" -> "number of moves".
- page 112: Theorem XV.8 is false the way you defined subrings.
Indeed, you have not required a subring to have the same unity as the
ring (just to have *some* unity); thus, a factor of a direct product
of two nontrivial rings is a subring of the direct product, but fails
condition (a) of Theorem XV.8.
- page 112, Example XV.10: "has an subrings" -> "has a subring
different from itself".
- page 113, Example XV.12: "subring" -> "a subring" (on the last line
of the example).
- page 114, Example XV.19: "there some" -> "there is some".
- page 116, Example XV.24: Remove the comma after "Although".
- page 121, proof of Theorem XVII.2: "is unit" -> "is a unit".
- page 121, proof of Theorem XVII.2: "$ab+km$ -> $ab-km$".
- page 123, §XVII.3: Capitalize the "we" on the first line of this
section. Also, require $m \geq 1$ (or rather $m \geq 2$, since you
forbid trivial rings; you might then need to "manually" set
$\varphi(1) = 1$).