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\title{Math 721\\Spring 2011\\Homework 1}
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\centerline{\it Due Friday January 21, 2011.}


\centerline{({\it You may assume that $R-$modules are unitary.})}


\centerline{}

\noindent 1. (5 pt) Let $R$ be a commutative ring with identity and $M$ some $R-$module. Show that (as $R-$modules) there is an isomorphism

\[
\text{Hom}_R(R,M)\cong M.
\]

\centerline{}

\noindent 2. Let $R$ be a commutative ring with 1, $M$ an $R-$module and $m\in M$.
\begin{itemize}
\item[a)] (5 pt) Show that $I_m=\{r\in R\vert rm=0\}$ is an ideal of $R$. (If $I_m\neq 0$ then we say that $m$ is a torsion element of $M$.)
\item[b)] (5 pt) If $R$ is a domain, then show that the set of torsion elements of $M$ forms a submodule of $M$ (and show that this may not be the case if $R$ is not a domain).
\end{itemize}

\centerline{} 



\noindent 3. (5 pt) Let $f:A\longrightarrow B$ be an $R-$module homomorphism. Show $f$ is an isomorphism if and only if there is an $R-$module homomorphism $g:B\longrightarrow A$ such that $fg=1_B$ and $gf=1_A$. Are both these conditions necessary?

\centerline{}

\noindent 4. (5 pt) Let $f:A\longrightarrow A$ be an $R-$module homomorphism such that $f(f(x))=f(x)$ for all $x\in A$. Show that $A\cong\text{ker}(f)\oplus\text{Im}(f)$. 

\centerline{}

\noindent 5. In this problem, we consider various $R-$module structures.
\begin{itemize}
\item[a)] (5 pt) Let $R=\mathbb{Z}[x]$ and $M=\mathbb{Z}[x]$. Find at least 2 {\it unitary} $R-$module structures on $M$.
\item[b)] (5 pt) Let $R=\mathbb{Q}$ and $M=\mathbb{Z}$. Show that there is no unitary $R-$module structure on $M$.
\end{itemize}

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