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If a finitely generated module $M$ injected in a free module is , then would the image of $M$ injected inbe a free finitely generated module.?

If a finitely generated module $M$ injected in a free module is $M$ injected in a free finitely generated module.

If a finitely generated module $M$ injected in a free module, then would the image of $M$ be a free finitely generated module?

Post Closed as "Not suitable for this site" by Steven Landsburg, abx, András Bátkai, Tom De Medts, R.P.
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Finitely generated module

If a finitely generated module $M$ injected in a free module is $M$ injected in a free finitely generated module.