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\begin{document}

\textbf{Math 6320, Assignment 5  \hfill Due: End of April}

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%\medskip {\large Name: }

\begin{enumerate}
\item
Let $A$ be a finitely generated abelian group, viewed as an $\bbZ$-module. Describe $A_{\fp}$ for each prime ideal $\fp$ in $\bbZ$. Use the structure theorem for abelian groups.

%\textbf{Solution:} 

\item
Let $R$ be a ring. Verify that for any $r$ in $R$, the ring $R[x]/(xr-1)$ is (canonically isomorphic) to the localization of $R$ at the multiplicatively closed subset $\{r^{i}\mid i\ \ge0\}$. 

%\textbf{Solution:} 

\item
Let $I$ be an ideal in $R$ and $M$ an $R$-module such that $M_{\fm}=0$ for each maximal ideal $\fm\supseteq I$. Prove that $IM=M$.

%\textbf{Solution:} 

\item 
Let $R$ be a ring and $M$ a faithful $R$-module; this means that $\mathrm{ann}_{R}\, M=(0)$. Prove that when $M$ is noetherian, as an $R$-module, the ring $R$ is noetherian.

%\textbf{Solution:} 

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\item 
Let $R$ be a Noetherian ring and $\vf\colon R\to R$ a surjective homomorphism of rings. Is $\vf$ an isomorphism?

%\textbf{Solution:} 

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\item Let $K$ be a field.
\begin{enumerate}
\item
Suppose $f(x)$ in $K[x]$ has positive degree. Prove that $K[x]$ is a finitely generated $K[f(x)]$-module. 
\item
Let $R$ be a subring of $K[x]$ that contains $K$. Prove that $R$ is Noetherian.
\item
Describe a non-noetherian subring of $K[x,y]$.
\end{enumerate}
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\item 
Suppose $K$ is not algebraically closed. Prove that each algebraic set in $K^{n}$ is the zero set of a single polynomial.

%\textbf{Solution:} 

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\item
Prove that the subset $V = \{(t,t^{2},...,t^{n})\mid t \in \bbC\}$ of $\bbC^{n}$ is algebraic.

%\textbf{Solution:}

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\item
Let $K$ be an algebraically closed field and $L$ an extension field. If polynomials $f_{1},\dots, f_{c}$ in $K[x_{1},...,x_{n}]$ have a
common root in $L^{n}$, prove they have a common root in $K^{n}$.

%\textbf{Solution:} 

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\item
Let $\fm$ be a maximal ideal of $\bbR[x, y]$ containing $x^{2} + y^{2} + 1$. What is the quotient $\bbR[x, y]/\fm$?

%\textbf{Solution:} 

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\end{enumerate}

\end{document}

