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\noindent MATH 720, Algebra I \\
Exercises 12\\
Due Wed 03 Dec

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\noindent
Throughout this homework set, let $R$ and $S$ be  commutative rings with identity, and let $M$ be a unitary $R$-module.

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\begin{exer}
Prove that the following conditions are equivalent:
\begin{enumerate}[(i)]
\item
$M$ is noetherian as an $R$-module,
\item
every submodule $N\subseteq M$ is noetherian as an $R$-module, and
\item
for every submodule $N\subseteq M$, the quotient $M/N$ is noetherian as an $R$-module.
\end{enumerate}
\end{exer}

\begin{exer}
Prove that the following conditions are equivalent:
\begin{enumerate}[(i)]
\item
$R$ is a noetherian ring,
\item
for every ideal $I\subseteq R$, the quotient $R/I$ is a noetherian ring, 
\item
for every $n\in\bbn_+$ the polynomial ring $R[x_1,\ldots,x_n]$ is a noetherian ring, and
\item
for some $n\in\bbn_+$ the polynomial ring $R[x_1,\ldots,x_n]$ is a noetherian ring.
\end{enumerate}
\end{exer}


\begin{exer}[Bonus]
Let $S\to R$ be a ring epimorphism.
Prove that $M$ is noetherian as an $S$-module if and only if $M$ is noetherian as an $R$-module.
\end{exer}

\begin{exer}[Bonus]
Consider an exact sequence of $R$-module homomorphisms
$$0\to M'\xrightarrow{f'} M\xrightarrow{f}M''\to 0.$$
Prove that $M$ is noetherian if and only if $M'$ and $M''$ are both noetherian.
\end{exer}




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