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\begin{document}


\begin{center}\begin{large}\textbf{Math 221 - Problem Set 4\\
Due Wednesday, Nov 6}\end{large}\end{center}



\pagestyle{empty}
All rings are commutative.

\begin{enumerate} 
\item Let $R$ be a ring and $M$ an $R$-module. Let $\mathcal{J}$ be the filtration $$M = M_0 \supset M_1 \supset \cdots .$$ 
\begin{enumerate}
\item Give an example to show that the function $\inn : M \to \text{gr}_{\mathcal{J}}M $ sending $m$ to $\inn(m)$ is not necessarily a group homomorphism.
\item Let $f,g \in M$. Suppose for each $i$, $f$ is in $M_i$ if and only if $g \in M_i$. Show that either $ \inn(f) + \inn(g) = \inn(f+g) $ or $\inn(f) + \inn(g) = 0$.
\item Suppose $M = R$ and that $\mathcal{J}$ is multiplicative, in the sense that $M_i M_j \subset M_{i+j}$. Show that either $\inn(f)\inn(g) = \inn(fg)$ or $\inn(f)\inn(g)=0$.
\end{enumerate}

\item Let $I \subset R$ be an ideal. Suppose $\gr_I R$ is an integral domain. Let $J = (f) \subset R$.

\begin{enumerate}
\item  Show that $\inn(J) \subset \gr_I R$ is generated by $\inn(f)$.
\item Give an example of a local ring $R$ with $I$ maximal and $f \in I$ such that $\inn(J)$ is not generated by $\inn(f)$.
\end{enumerate}

\item Suppose $J \subset I$ are ideals in a ring $R$. Show that $$\gr_I (R/J) = (\gr_I R)/\inn(J).$$

\item Let $R$ be a Noetherian ring and $I \subset R$ an ideal. Let $M$ be a finitely generated $R$-module.

\begin{enumerate}
\item Show there is a largest submodule $N \subset M$ such that $N$ is annihilated by an element of the form $1-r$ with $r \in I$.
\item Show that $\bigcap_{j = 1}^\infty I^j M = N.$
\end{enumerate}


\end{enumerate}
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