Let $R$ be a ring, let $M$ and $N$ be left $R$-modules, and let
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\begin{align*}
f: M \to N
\end{align*}
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be an $R$-[module homomorphism](/page/Module%20Homomorphism). If $M$ is finitely generated as a left $R$-module, then $\operatorname{im} f$ is finitely generated as a left $R$-module.