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\title{Math 122: Homework 1}
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\begin{itemize}[noitemsep]
  \item Section 10.1 \# 7,8a,18,19
  \item Section 10.3 \# 9,10,11
\end{itemize}

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{\textbf{Section 10.1 \# 7}}. Let $N_1 \subseteq N_2 \subseteq \cdots$ be an
ascending chain of submodules of $M$. Prove that $\bigcup_{i = 1}^{\infty}
N_i$ is a submodule of $M$.

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{\textbf{Section 10.1 \# 8a}}. An element $m$ of the R-module $M$ is called
a {\textit{torsion element}} if $r m = 0$ for some nonzero element $r \in R$.
The set of torsion elements is denoted\footnote{This notation is arguably
objectionable since if $M, N$ are modules $\operatorname{Tor} (M, N)$ has a different
meaning in homological algebra.}
\[ \operatorname{Tor} (M) = \left\{ m \in M| \text{$r m = 0$ for some nonzero $r \in
   R$} \right\} . \]


(a) Prove that if $R$ is an integral domain then $\operatorname{Tor} (M)$ is a
submodule of $M$ (called the {\textit{torsion submodule}} of $M$).

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{\textbf{Section 10.1 \# 18}}. Let $F =\mathbb{R}$, let $V=\mathbb{R}^2$ and
let $T$ be the linear transformation from $V$ to $V$ which is rotation
clockwise around the origin by $\pi / 2$ radians. Show that $V$ and $0$ are the
only $F [x]$-submodules for this $T$.

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{\textbf{Section 10.1 \# 19}}. Let $F =\mathbb{R}$, let $V =\mathbb{R}^2$
and let $T$ be the linear transformation from $V$ to $V$ which is projection
on the $y$-axis. Show that $V, 0$, the $x$-axis and the $y$-axis are the only
$F [x]$-submodules of $V$ for this $T$.

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{\textbf{Section 10.3 \# 9.}} An $R$-module $M$ is {\textit{irreducible}} if
$M \neq 0$ and if $0$ and $M$ are the only submodules of $M$. Show that $M$ is
irreducible if and only if $M \neq 0$ and $M$ is a cyclic module with any
nonzero element as a generator. Determine all the irreducible
$\mathbb{Z}$-modules.

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{\textbf{Section 10.3 \#10}}. Assume that $R$ is commutative. Show that an
$R$-module $M$ is irreducible if and only if $M$ is isomorphic (as an
$R$-module) to $R / I$ where $I$ is a maximal ideal of $R$.
[{\textbf{Hint:}} By the previous exercise, if $M$ is irreducible there is a
natural map $R \rightarrow M$ defined by $r \mapsto r m$, whree $m$ is any
fixed nonzero element of $M$.]

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{\textbf{Section 10.3 \#11}}. Show that if $M_1$ and $M_2$ are irreducible
$R$-modules, then any nonzero R-module homomorphism from $M_1$ to $M_2$ is an
isomorphism. Deduce that if $M$ is irreducible then $\operatorname{End}_R (M)$ is a
division ring. (This result is called {\textit{Schur's Lemma}}).
[{\textbf{Hint:}} consider the kernel and image.]

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