\begin{align*}
0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0
\end{align*}
latex_env
admin
be a short exact sequence of left $R$-modules. If $A$ and $C$ are finitely generated as left $R$-modules, then $B$ is finitely generated as a left $R$-module. Moreover, if $B$ is finitely generated as a left $R$-module, then $C$ is finitely generated as a left $R$-module.