Math 6310 • Algebra

Let \(R\) be a commutative ring with identity, and let \(0\to L\to M\to N\to0\) be an exact sequence of \(R\)-
modules. Prove or disprove:
\begin{enumerate}

\item
If \(M\) and \(N\) are finitely generated \(R\)-modules, then \(L\) is finitely generated.


\item
If \(M\) is finitely generated, and \(N\) is a free \(R\)-module, then \(L\) is finitely generated.

\end{enumerate}