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\begin{center}\begin{large}\textbf{Math 523 - Problem Set 3\\
Due Wednesday, March 2}\end{large}\end{center}



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\begin{enumerate}

\item Let $k = \Z/(2)$ and $R = k[x,y]/(x,y)^2$. Show that $(x,y)$ is the union of three smaller ideals. Why doesn't this contradict prime avoidance?

\item Let $k$ be an infinite field. Show that $(x,y)$ in $k[x,y]$ is contained in an infinite union of primes $P_i$ such that no $P_i$ contains $(x,y)$. Why doesn't this contradict prime avoidance?

\item Let $M$ be a finitely generated $\Z$-module (i.e. abelian group). Describe the set of associated primes of $M$ in terms of the usual structure theory for finitely generated abelian groups.

\item Let $R = \C[x,y,z]/(x^2-yz)$. Let $X, Y, Z$ be the images of $x,y,z$ in $R$. Find minimal primary decompositions of $(X)$ and $(Y)$ in $R$. In each case, list the corresponding associated primes.

\item Let $R = \C[x,y,z]$. Let $I = (x^2yz, z^2)$. Find the support and associated primes of $R/I$, and give a minimal primary decomposition of $I$.

\item Let $R$ be a ring and $A$ an $n \times n$ matrix with entries in $R$, $\text{adj}(A)$ the adjugate matrix of $A$, and $I_{n\times n}$ the $n \times n$ identity matrix. In this exercise we will prove the following matrix equation (which is a fact from linear algebra when $R$ is a field): $$ \text{adj}(A) A = \det(A) I_{n\times n}.$$ 
\begin{enumerate}
\item Show that $R$ is isomorphic to the quotient of a polynomial ring $S = \Z[x_1, x_2, \ldots]$ (in possibly infinitely many variables).
\item Show that to prove the matrix equation for $R$ it suffices to prove it for $S$.
\item Show that in order to prove the matrix equation over $S$, it suffices to prove it over its fraction field, and conclude the equation from the known linear algebra fact.
\end{enumerate}

\item Let $k$ be a field and $R = k[t]/(t^2)$. Set $$ p(x) = tx^3+tx^2-x^2-x \in R[x].$$ 

\begin{enumerate}
\item Show that $S = R[x]/(p)$ is a free $R$ module of rank 2.
\item We will prove a theorem in class that says $S$ is a free module if and only if $(p)$ is generated by a monic polynomial. Clearly $p$ is not monic. How do you reconcile these two facts?

\end{enumerate}

\item Let $R$ be a local ring, with maximal ideal $m$. Let $I \subset R$ be an ideal, and suppose $x \in m$ is a nonzerodivisor on $R/I$. 

\begin{enumerate}
\item Show that if $I$ has a minimal set of generators, then it maps to a minimal generating set for the image of $I$ in $R/(x)$.\footnote{Hint: Wait to do this problem until we've stated Nakayama's Lemma.}
\item Give an example to show that (a) can fail if $x$ is a zerodivisor on $R/I$.
\end{enumerate}

\item \textbf{{(Fun/optional:)}} Let $k$ be a field and $R = k[x_1, \ldots, x_n]$. An ideal $I \subseteq R$ is a \textbf{\textit{monomial ideal}} if it can be generated by monomials (i.e. elements of the form $x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$, for $a_i \geq 0$). Which monomial ideals are prime? primary? radical?
 
\end{enumerate}



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