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\begin{center}\begin{large}\textbf{Math 221 - Problem Set 1\\
Due Wednesday, Sept 25}\end{large}\end{center}



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All rings are commutative.

\begin{enumerate} 

\item Prove that the following conditions on a module $M$ over a ring $R$ are equivalent.

\begin{enumerate}[(i)]
\item $M$ is Noetherian.
\item Every ascending chain of submodules terminates.
\item Every set of submodules of $M$ contains elements that are maximal under inclusion.
\item Given a sequence of elements $f_1, f_2, \ldots \in M$, there is some $m$ such that for $n >m$, there is an expression $$f_n = \sum_{i=1}^{m} a_i f_i,$$ with $a_i \in R$.
\end{enumerate}

\item Let $R$ be a Noetherian ring, and $I \subseteq R$ an ideal. A prime $P$ containing $I$ is \textit{\textbf{minimal over $I$}} if there is no prime $Q$ such that $I \subset Q \subsetneq P$. Show that there are only finitely many primes minimal over $I$ . \textit{[Hint: Assume not. Then there is an ideal $I$ maximal among ideals for which it fails. This ideal can't be prime, so there are $f,g \in R - I$ such that $fg \in I$. Show that any prime maximal over $I$ is maximal over $(I, f)$ or $(I,g)$.]}

\item Let $k$ be a field. Compute the Hilbert function and polynomial for $k[x,y,z,w,]/(x,y)\cap(z,w)$.

\item Let $R$ be a ring, $I \subset R$ an ideal, and $f \in R$. We define $$(I : f) := \{r\in R \mid fr \in I\}.$$

\begin{enumerate}
\item Check that $(I:f)$ is an ideal.
\item Let $U \subset R$ be a multiplicative set. Show that there is a one-to-one correspondence between the ideals in $R[U^{-1}]$ and ideals $I$ in $R$ such that $(I:f)= I$ for all $f \in U$. \
\item Show that this correspondence sends prime ideals to prime ideals.
\item Show that for any ideal $J \subset R$, we have $R \cap JR[U^{-1}] = \sum_{f\in U}(J:f)$.
\item Show that the image of the map $J \mapsto R \cap JR[U^{-1}]$ is exactly the set of ideals $I \subset R$ such that $(I:f) = I$.
\end{enumerate}

\item Let $k$ be a field and $m,n$ positive integers. Describe as explicitly as possible each of the following modules.

\begin{enumerate}
\item $\Hom_\Z (\Z/(n),\Z/(m))$
\item $\Hom_{k[x]} (k[x]/(x^n),k[x]/(x^m))$
\item $\Z/(n) \otimes_\Z \Z/(m)$
\item $k[x]/(x^n) \otimes_{k[x]} k[x]/(x^m)$
\end{enumerate}

\item Show that the universal property of localization that we stated in class characterizes $R \to R[U^{-1}]$ up to unique isomorphism. That is if another map $R \to S$ satisfies the universal property, then there is a unique isomorphism $R[U^{-1}] \to S$ such that 
$$\xymatrix@=10pt{& R \ar[dl] \ar[dr] &  \\ R[U^{-1}] \ar[rr] && S } $$
commutes.
\end{enumerate}
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