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



\pagestyle{empty}
All rings are commutative.

\begin{enumerate} 

\item Characterize the $p$-adic expansions in $\Z_p$ which are images of integers.

\item \begin{enumerate} 
\item Let $R$ be a Noetherian ring. If $R$ is local, and the maximal ideal $\mathfrak{m}$ is principal, show that every nonzero ideal of $R$ is a power of $\mathfrak{m}$.
\item Deduce, without using the Principal Ideal Theorem, that if each maximal ideal of a Noetherian ring $R$ is principal, $\dim R \leq 1$.
\end{enumerate}

\item Let $R \to S$ be a homomorphism of rings, so that $S$ is integral over $R$. Let $M$ be an $S$-module. Show that the dimension of $M$ as an $S$-module is equal to the dimension of $M$ as an $R$-module.

\item Let $R$ be a ring and $U \subset R$ multiplicatively closed. Define $S = R[U^{-1}]$. Show that for any prime $P \subset S$, the codimension of $P$ in $S$ is equal to the codimension of   $P \cap R$ in $R$.

\item Let $k$ be a field.
\begin{enumerate}
\item Show that $\phi: k[x',y] \to k[x, y]$ defined $\phi(x') = x-y^n$ and $\phi(y) = y$ is an isomorphism of $k$-algebras.
\item Let $f \in k[x,y]$. Show that if $n >>0$, then $\phi^{-1}(f) \in k[x',y]$ is a scalar times a monic polynomial in $y$ over $k[x']$. \textit{[Hint: the $n$ you choose will depend on $f$. i.e. you can't choose $f$ to be $x-y^n$.]}
\item Deduce that $k[x,y]/(f)$ is integral over the subring $\phi(k[x'])$.
\item Use (c) to prove that $\dim k[x,y] = 2.$
\end{enumerate}

\item Let $R$ be a Noetherian ring. Show that $\dim R[x, x^{-1}] = 1 + \dim R$.

\end{enumerate}
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