# Tagged: finitely generated module

## Problem 414

Let $R$ be a ring with $1$. Let
$0\to M\xrightarrow{f} M’ \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}$ be an exact sequence of left $R$-modules.

Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M’$ is also finitely generated.