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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 1\\
Due Wednesday, Feb 5}\end{large}\end{center}



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All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

\item Let $R = k[x_1, \ldots , x_n]$. Verify the following statements from class.

\begin{enumerate}
\item Let $\{I_\alpha\}$ be a collection of ideals in $R$. Show that $\bigcap_\alpha V(I_\alpha) = V(\cup_\alpha I_\alpha).$
\item Let $I, J \subset R$ be ideals. Show that $V(IJ) = V(I) \cup V(J).$
\end{enumerate}

\item Let $J \subset k[x_1, \ldots, x_n]$ be an ideal and $X$ and $Y$ algebraic sets. Verify the following statements.

\begin{enumerate}
\item $V(I(V(J))) = V(J)$.
\item $I(V(I(X))) = I(X)$.
\item $X = Y$ if and only if $I(X) = I(Y)$.
\end{enumerate}

\item Let $f \in k[x,y]$ be a polynomial of degree $n >0$, and $C = V(f)$. Let $L$ be a line in $\A^2_k$ such that $L$ is not contained in $C$. Show that $L \cap C$ is a finite set of no more than $n$ points.

\item Determine (and prove) whether each of the following is an algebraic set.

\begin{enumerate}
\item $\{(t,t^2,t^3) \in \A_k^3 \mid t\in k\}$.
\item The set of $m \times n$ matrices over $k$ with rank $\leq r$.
 respectively.
\item The closed unit ball $\{P \in  \A^n_{\C} \mid \| P \| \leq 1\}$.
\item The set $U \times V \in \A^{m+n}$, where $U$ and $V$ are algebraic sets in $\A^m$ and $\A^n$, respectively.
\end{enumerate}


\item Let $R$ be a ring. Show that the following statements are equivalent. (Hint: You will need to use Zorn's Lemma.)
\begin{enumerate}[(i)]
\item $R$ is Noetherian.
\item Every strictly increasing sequence of ideals $I_1 \subsetneq I_2 \subsetneq \ldots $ is finite.
\item Every set of ideals $\mathcal{J}$ of $R$ contains an element $I$ such that no other element of $\mathcal{J}$ contains $I$.
\end{enumerate}

\end{enumerate}
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