% LaTeX solution template for Math 332
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%	VARIOUS USEFUL COMMANDS
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% The commands below might be convenient. For example, you probably
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%	MAKING SUMMATION SIGNS ALWAYS PUT THEIR BOUNDS ABOVE AND BELOW
%	THE SIGN
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%	METADATA
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\newcommand{\myname}{Darij Grinberg} % ENTER YOUR NAME HERE
\newcommand{\mymail}{darij.grinberg@drexel.edu} % ENTER YOUR EMAIL HERE
\newcommand{\psetnumber}{5} % ENTER THE NUMBER OF THIS PSET HERE

%----------------------------------------------------------------------------------------
%	HEADER AND FOOTER
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\ihead{Solutions to homework set \#\psetnumber} % Page header left
\ohead{page \thepage\ of \pageref{LastPage}} % Page header right
\ifoot{\myname} % left footer
\ofoot{\mymail} % right footer

%----------------------------------------------------------------------------------------
%	TITLE SECTION
%----------------------------------------------------------------------------------------
\title{	
\normalfont \normalsize 
\textsc{Drexel University, Department of Mathematics} \\ [15pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Math 332: Undergraduate Abstract Algebra II, \\
Winter 2025:
Homework \psetnumber\\% The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
Please solve \textbf{at most 3 of the 6 problems!}
% \footnote{%
% I recommend solving as many problems as you can and wish,
% but I will only grade and score 3 solutions per submission
% (and if you submit more, I get to pick which ones I grade).
% \\
% Results stated in class, and claims of previous problems
% (even if you did not solve these previous problems), can
% be used without proof. For example, in solving Problem 5,
% you can use the result of Problem 2 without proof.
% \\
% I expect approximately the level of detail that I give in class.
% Purely straightforward arguments (like checking the ring
% axioms for a direct product of rings) need not be spelled out;
% I only expect a note of their necessity.
% }
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
\author{\myname}

\begin{document}

\maketitle % This command causes the title part to be printed.

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 1}

\subsection{Problem}

Let $R$ be a ring.
Let $I$, $J$ and $K$ be three mutually comaximal ideals of $R$.
Prove that $IJ + JK + KI = R$.

\subsection{Hint}

Expand
$\tup{i_1+j_1}\tup{j_2+k_2}\tup{k_3+i_3}\tup{i_4+j_4}\tup{j_5+k_5}$.

Keep in mind that $R$ need not be commutative!

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 2}

\subsection{Problem}

Prove that the ring
\[
\ZZ \ive{\sqrt{-2}}
:= \set{ a + b \sqrt{-2} \ \mid \ a, b \in \ZZ }
\]
is Euclidean, and that the map
\begin{align*}
N : \ZZ \ive{\sqrt{-2}} &\to      \NN, \\
    a + b \sqrt{-2}     &\mapsto  a^2 + 2b^2
\qquad
\tup{\text{for $a, b \in \ZZ$}}
\end{align*}
is a Euclidean norm for it.

\subsection{Hint}

Imitate the proof for $\ZZ\ive{i}$ in \cite[\S 2.13.2]{notes}.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 3}

\subsection{Problem}

Let $p$ be a prime such that $p \equiv 1 \mod 4$.
Prove that $p$ can be written in the form
$p = x^2 + 4y^2$ for some $x, y \in \ZZ$.

\subsection{Hint}

No new rings are required!

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 4}

\subsection{Problem}

Let $R$ be the ring $\ZZ\ive{i}$ of Gaussian integers.
Let $S$ be the ring
\begin{align*}
\ZZ\ive{2i}  &= \set{ a + b \cdot 2i \ \mid \ a, b \in \ZZ } \\
&= \set{\text{Gaussian integers with an even imaginary part}}  .
\end{align*}
This ring $S$ is a subring of $R$.

Define two elements $x, y \in S$ by $x = 2+2i$ and $y = 2-2i$.

\begin{enumerate}

\item[\textbf{(a)}]
Find the units of $S$.

\item[\textbf{(b)}]
Prove that we have $x \sim y$ in $R$, but we don't have
$x \sim y$ in $S$.

\item[\textbf{(c)}]
Prove that the ideal $xS + yS$ of $S$ is not principal.

\item[\textbf{(d)}]
Conclude that $S$ is not a PID.

\end{enumerate}

\subsection{Hint}

It may be helpful to write $i'$ for $2i$ in order to
avoid confusing $i$ for an element of $S$.

For part \textbf{(c)}, argue that if $xS + yS$ was $zS$ for some
$z \in S$, then $xR + yR$ would be $zR$ as well (why?), but this
would force $z$ to be associate to $x$ and $y$ in $R$ (why?),
and this would leave only four possibilities for $z$ (why?).

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 5}

\subsection{Problem}

Let $R$ be a ring, and $n \in \NN$.
Consider the left $R$-module $R^n$.

\begin{enumerate}

\item[\textbf{(a)}]
Prove that the set
\begin{align*}
A := &  \set{ \tup{a_1, a_2, \ldots, a_n} \in R^n
              \ \mid \ a_1 = a_2 = \cdots = a_n } \\
   = & \set{ \tup{\underbrace{r,r,\ldots,r}_{n\text{ times}}}
              \ \mid \ r \in R }
\end{align*}
is an $R$-submodule of $R^n$.

\item[\textbf{(b)}]
Prove that the set
\[
B :=  \set{ \tup{a_1, a_2, \ldots, a_n} \in R^n
            \ \mid \ 1 a_1 = 2 a_2 = \cdots = n a_n }
\]
is an $R$-submodule of $R^n$.

\item[\textbf{(c)}]
Prove that the set
\[
C := \set{ \tup{a_1, a_2, \ldots, a_n} \in R^n
           \ \mid \ a_1 a_2 \cdots a_n = 0 }
\]
is an $R$-submodule of $R^n$ \textbf{only if}
$R$ is trivial or $n=1$.

\item[\textbf{(d)}]
Prove that the set
\[
D := \set{ \tup{a_1, a_2, \ldots, a_n} \in R^n
           \ \mid \ a_i = a_{i-1} + a_{i-2} \text{ for all }
                        i \geq 3 }
\]
is an $R$-submodule of $R^n$.

\end{enumerate}

\subsection{Hint}

For the sake of brevity, feel free to abbreviate an $n$-tuple
$\tup{a_1, a_2, \ldots, a_n}$ as $a$.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 6
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 6}

\subsection{Problem}

Let $R$ be a ring. Let $M$ be a left $R$-module.
Let $I$ be an $R$-submodule of $M$.

For any two elements $a, b \in M$, we write
``$a \equiv b \mod I$'' (and say that ``$a$ is congruent to
$b$ modulo $I$'') if and only if $a - b \in I$.
(This is a generalization of congruence of integers, as it is
usually defined in elementary number theory.
Indeed, congruence of integers modulo an integer
$n$ is recovered when $R = \ZZ$ and $I = n \ZZ$.)

Prove the following:

\begin{enumerate}

\item[\textbf{(a)}]
Each $a \in M$ satisfies $a \equiv a \mod I$.

\item[\textbf{(b)}]
If $a, b \in M$ satisfy $a \equiv b \mod I$, then
$b \equiv a \mod I$.

\item[\textbf{(c)}]
If $a, b, c \in M$ satisfy $a \equiv b \mod I$ and
$b \equiv c \mod I$, then $a \equiv c \mod I$.

\item[\textbf{(d)}]
If $a, b, c, d \in M$ satisfy $a \equiv b \mod I$ and
$c \equiv d \mod I$, then $a+c \equiv b+d \mod I$.

\item[\textbf{(e)}]
If $a, b\in M$ and $r \in R$ satisfy
$a \equiv b \mod I$, then $ra \equiv rb \mod I$.

\end{enumerate}

Now, we claim a sort of converse:

\begin{enumerate}

\item[\textbf{(f)}]
Let us drop the requirement that $I$ be an
$R$-submodule of $M$.
Instead, we require that the claims of parts
\textbf{(a)}, \textbf{(c)} and \textbf{(e)} of this
exercise hold.
Prove that $I$ is an $R$-submodule of $M$.

\end{enumerate}

\subsection{Hint}

This exercise can be summarized as ``modular arithmetic
modulo a subset $I$ of $M$ works if and only if $I$ is
a submodule of $M$''. In other words, roughly speaking,
the submodules of a module $M$ are precisely the
subsets $I$ that allow ``working modulo $I$''.
This is most likely the reason why modules are called
``modules''\footnote{The name was coined by Dedekind,
although in a less general context.}.

\subsection{Solution}

...


\horrule{0.3pt} \\[0.4cm]

\begin{thebibliography}{99999999}

% \bibitem[DF]{DF}
% David S. Dummit, Richard M. Foote,
% \textit{Abstract Algebra},
% 3rd edition,
% Wiley 2004.

\bibitem[23wa]{notes}
Darij Grinberg,
\textit{Math 332 Winter 2023 notes: An introduction
to the algebra of rings and fields},
17 January 2025.
\url{https://www.cip.ifi.lmu.de/~grinberg/t/23wa/23wa.pdf}

\end{thebibliography}

\end{document}

