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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 4.2  \\ Fall 2019 \hfill Due October 2}}}

\large

\vspace{1cm}

\begin{exercise}
Prove that a commutative ring $R$ is a domain if and only if $(0)$ is a prime ideal.
\end{exercise}

\begin{exercise}
A ring $R$ is called {\em local} if it has a unique maximal ideal.  Prove that $R$ is a local ring with maximal ideal $M$ if and only if $M = R \setminus R^{\times}$.
\end{exercise}

\begin{exercise}
The {\em spectrum} $\operatorname{Spec} R$ of a commutative ring $R$ is defined to be the set of all prime ideals in $R$.  Given an ideal $A \subset R$, the {\em ideal variety} in $\operatorname{Spec} R$ associated to $A$ is
\[
V(A) = \{ P \in \operatorname{Spec} R \, | \, A \subset P \}.
\]
Let $B \subset R$ be an ideal.

\begin{itemize}

\item[\bf a.]  Prove that $A \subset B$ implies $V(B) \subset V(A)$.  

\item[\bf b.]  Prove that $V(A + B) = V(A) \cap V(B)$.  

\item[\bf c.]  Prove that $V(AB) = V(A) \cup V(B)$.

\end{itemize}

\noindent{\bf Remark.}  Part {\bf b} can be generalized as follows.  If $\mathcal{I}$ is any collection of ideals, and we define the sum of the ideals in $\mathcal{I}$ by
\[
\sum_{A \in \mathcal{I}} A = \left( \bigcup_{A \in \mathcal{I}} A \right) , 
\]
we leave it  as an additional exercise to show that $V(\sum_{A \in \mathcal{I}} A ) = \bigcap_{A \in \mathcal{I}} V(A)$.  Together with part {\bf c} this proves that the ideal varieties $V(I)$ are the closed sets of a topology on $\operatorname{Spec} R$, known as the {\em Zariski topology}.  In modern algebraic geometry, the space $\operatorname{Spec} R$ equipped with the Zariski topology (and some additional algebraic information) is called an {\em affine scheme}.


\end{exercise}

\begin{exercise}
Let $I$ be an interval of real numbers and $R = C^0(I)$ be the set of all continuous real-valued functions on $I$.  Note that $R$ is a subring of $\R^{I}$ by standard results from calculus.  Prove that for each $a \in I$, the set
\[
M_a = \{ f \in R \, | \, f(a)=0 \}
\]
is a maximal ideal in $R$.  Are these the only maximal ideals in $R$?
\end{exercise}

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