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\title{Math 332: Homework 10}
%\author{Your name here --- delete the % at the start of the line}
\date{20.IV.15}

\begin{document}
\maketitle

Throughout, $R$ is a (not necessarily commutative) ring with $1$ unless otherwise specified.

\begin{exc}
Suppose $M$ is a left $R$-module.  Prove that $0m=0$ and $-1m=-m$ for all $m\in M$.
\end{exc}

\begin{exc}
Let $F = \RR$, let $V=\RR^2$, and let $T$ be the linear transformation $V\to V$ which is projection onto the second coordinate.  Show that $V$, $0$, $\RR\times 0$, and $0\times \RR$ are the only $F[x]$-submodules of $V$ for this choice of $T$.
\end{exc}

\begin{prob}
If $N$ is a submodule of an $R$-module $M$, the \emph{annihilator} of $N$ in $R$ is defined to be
\[
  \Ann(N) = \{r\in R\mid rn=0\text{ for all }n\in N\}.
\]
Prove that $\Ann(N)$ is a 2-sided ideal of $R$.
\end{prob}

\begin{prob}
Let $A$ be any $\ZZ$-module, let $a$ be any element of $A$, and let $n$ be any positive integer.
\begin{enumerate}[(a)]
\item Prove that $\varphi_a:\ZZ/n\ZZ\to A$ given by $\varphi(\bar{k}) = ka$ is a well-defined $\ZZ$-module homomorphism if and only if $na=0$.
\item Prove that $\Hom_\ZZ(\ZZ/n\ZZ,A)\cong {}_n A$, where ${}_n A = \{a\in A\mid na=0\}$.
\end{enumerate}
\end{prob}

\begin{prob}
Prove that $\Hom_\ZZ(\ZZ/n\ZZ/m\ZZ)\cong \ZZ/(n,m)\ZZ$.
\end{prob}

\begin{chal}
Assume $R$ is commutative.  Prove that $R^n\cong R^m$ if and only if $n=m$, \emph{i.e.}, two free $R$-modules of the same rank are isomorphic if and only if they have the same rank.  (See Exercise 2 on p.356 for a hint.)
\end{chal}

\begin{prob}
Let $N$ be a submodule of $M$.  Prove that if both $M/N$ and $N$ are finitely generated, then so is $M$.
\end{prob}

\begin{prob}
Determine the dimension of $\CC\otimes_\RR \CC$ as a $\CC$-vector space.  Determine the natural ring structure on $\CC\otimes_\RR \CC$.
\end{prob}


\end{document}