If a finitely generated module $M$ injected in a free module, then would the image of $M$ be a free finitely generated module?
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$\begingroup$ I meet that question while I was testing the coherence in the case of representation theory of quivers. I think that my question was so interesting. $\endgroup$– Syzygy AdnanCommented Sep 19, 2019 at 21:10
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If a finitely generated $M$ is mapped into the free module on some set $X$, then the image of each generator of $M$ is a linear combination of finitely many elements of $X$. The images of the finitely many generators of $M$ involve, in this way, only finitely many elements of $X$ altogether. So the given map sends $M$ into the submodule freely generated by those finitely many elements of $X$.
answered Sep 19, 2019 at 0:07
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$\begingroup$ Firstly i would like to thank you for the answer. I meet that question while I was testing the coherence in the case of representation theory of quivers. $\endgroup$ Commented Sep 19, 2019 at 21:12