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\begin{center}\begin{large}\textbf{Math 221 - Problem Set 2\\
Due Wednesday, Oct 9}\end{large}\end{center}



\pagestyle{empty}
All rings are commutative.

\begin{enumerate} 

\item Let $R$ be a ring, $I \subset R$ an ideal, and $U \subset R$ a multiplicative set.
\begin{enumerate}
\item Show that the natural map $f: \Spec(R/I) \to \Spec(R)$ is a homeomorphism onto its image $V(I)$. That is, check that it is invertible, and that both $f$ and $f^{-1}$ send closed sets to closed sets.\footnote{The closed sets of $X \subset \Spec(R)$ are those of the form $V(J) \cap X$ for $J$ an ideal in $R$.}
\item Show that the natural map $f: \Spec(R[U^{-1}]) \to \Spec(R)$ is a homeomorphism onto its image $Y =\{P \mid U \cap P = \emptyset\}$.
\end{enumerate}

\item Let $R$ be a ring, $M$ an $R$-module. Define the \textbf{\textit{support}} of $M$, $\Supp (M) \subset \Spec(R)$, to be the set of prime ideals $P$ such that $M_P \neq 0$. 

\begin{enumerate}
\item Show that if $M$ is finitely generated, $\Supp (M)$ is the set of prime ideals containing the annihilator of $M$, and thus it is closed.

\item Let $R = k[x,y,z]$, $k$ algebraically closed. Let $I = (x^2yz, z^2) \subset R$. Find the support and associated primes of $R/I$, and write $I$ as the intersection of primary ideals.
\end{enumerate} 

\item Let $R$ be a ring and $M$ an $R$-module. Let $\{f_i\}$ be a set of elements of $R$ that generate the unit ideal. 

\begin{enumerate}
\item Show that if $m \in M$ goes to 0 in each $M[f_i^{-1}],$ then $m=0$. 
\item Show that if $m_i \in M[f_i^{-1}]$ are elements such that $m_i$ and $m_j$ go to the same element of $M[f_i^{-1}f_j^{-1}],$ then there is a unique element $m \in M$ such that $m$ goes to $m_i$ in $m_i \in M[f_i^{-1}]$ for each $i$. 
\end{enumerate}

\item Let $k$ be a field. A  \textbf{\textit{monomial ideal}} is an ideal $I \subset k[x_0, x_1, \ldots, x_r]$ generated by monomials. Which monomial ideals are prime? Irreducible? Radical? Primary?

\item Here are some examples to show that prime avoidance can't be improved. \begin{enumerate}

\item Show that if $k = \Z/(2)$, then the ideal $(x,y) \subset k[x,y]/(x,y)^2$ is the union of three properly smaller ideals.
\item Let $k$ be any field. In the ring $k[x,y]/(xy,y^2)$, consider the ideals $I_1 = (x)$, $I_2 = (y)$, and $J = (x^2, y)$. Show that the homogeneous elements of $J$ are contained in $I_1 \cup I_2$, but $J$ is not contained in either.

\end{enumerate}

\item Show that if $k$ is an infinite field, the ideal $(x,y)$ in $k[x,y]$ is contained in an infinite union of primes $P_i$ such that no $P_i$ contains $(x,y)$. That is, prime avoidance can fail for an infinite number of primes.

\end{enumerate}
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