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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Weekly Homework \#4}
\date{Assigned Monday, 16 October 2017; due Monday, 23 October 2017}

\maketitle

If not otherwise specified, let $R$ be a ring (with $1$) and let $M$ be a (left) $R$-module.  

\vspace{1ex}

\begin{problab}{W4.1}
Let $R$ be a (commutative integral) domain.

\begin{enumalph}
\item Let $k$ be a field and suppose that $R$ is a finite-dimensional $k$-algebra.  Show that $R$ is a field.  
\item Show that if $R$ is finite ($\#R < \infty$), then $R$ is a field.
\end{enumalph}
\end{problab}

\begin{problab}{W4.2}
Let $N$ be an $R$-submodule of $M$.  Prove that if both $M/N$ and $N$ are finitely generated, then so is $M$.\end{problab}

\begin{problab}{W4.3}
$M$ is called \emph{irreducible} if $M \neq \{0\}$ and the only $R$-submodules of $M$ are $\{0\}$ and $M$.  
\begin{enumalph}
\item Show that $M$ is irreducible if and only if $M \neq \{0\}$ and $M$ is a cyclic module with any nonzero element as generator.
\item Determine all the irreducible $\Z$-modules.
\item Suppose that $R$ is commutative.  Show that $M$ is irreducible if and only if $M \simeq R/J$ where $J$ is a maximal ideal of $R$.
\item Show (again over an arbitrary ring $R$) that if $M_1,M_2$ are irreducible $R$-modules, then any $R$-module homomorphism $M_1 \to M_2$ is either zero or an isomorphism.  Deduce that if $M$ is irreducible then $\End_R(M)$ is a division ring.
\end{enumalph}
\end{problab}


\begin{problab}{W4.4}
Suppose $R$ is commutative.  
\begin{enumalph}
\item Let $M_i$ be $R$-modules for $i \in I$ and let $N$ be another $R$-module.  Show that 
\[ \Hom_R\biggl(\bigoplus_{i\in I} M_i, N\biggr) \simeq \prod_{i \in I} \Hom_R(M_i,N) \] 
are isomorphic as $R$-modules. 
\item Let $N_i$ be $R$-modules for $i \in I$.  Show that there is an injective map  
\[ \bigoplus_{i \in I} \Hom_R(M,N_i) \hookrightarrow \Hom_R\biggl(M,\bigoplus_{i\in I} N_i\biggr)  \] 
of $R$-modules that is an isomorphism when either $\# I<\infty$ or when $M$ is finitely generated.  
\end{enumalph}
\emph{[Hint: Make a big deal about why it is the direct product in one case and the direct sum in another.  Of course there is only a distinction when $\# I = \infty$.]}
\end{problab}

\begin{problab}{W4.5}
Let $R=\Z[x]$ be the ring of polynomials in a variable $x$ with integer coefficients.  Let $I=(2,x)$ be the ideal generated by $2$ and $x$ in $R$.  
\begin{enumalph}
\item Show that $R/I \simeq \Z/2\Z$ as rings and as $R$-modules.
\item Show that $2 \otimes 2 + x \otimes x \in I \otimes_R I$ is nonzero and cannot be written as a simple tensor (in the form $a \otimes b$ for some $a,b \in I$).  
\item The element $\alpha = 2 \otimes x - x \otimes 2 \in I \otimes_R I$ is nonzero.  Compute its annihilator $\ann(\alpha) = \{r \in R : r\alpha =0\}$ as an ideal in $R$.  Show that the $R$-submodule $R\alpha \subseteq I \otimes_R I$ generated by $\alpha$ is isomorphic as an $R$-module to $R/I$.   
\end{enumalph}
\end{problab}

\end{document}