Consider the folowing property for an $R$-module $M$ (where $R$ is an associative ring with identity):
For each family $\{M_{\alpha}\}_{\alpha\in I}$ of $R$-modules and each homomorphism $f:M\to \bigoplus_{\alpha\in I}M_{\alpha}$, there exists a finite subset $J$ of $I$ such that $\text{Im}f\subseteq\bigoplus_{\alpha\in J}M_{\alpha}$.
Every finitely generated $R$-module $M$ satisfies the condition.
My QUESTION is:
Is this condition a characterization of finitely generated $R$-modules, or is there an example of an $R$-module which satisfies the property but it is not finitely generated?