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Suppose $R=k[x_1,...,x_n]$ is a polynomial ring over a field (I'm only interested in the case when $k$ has characteristic zero, but I don't know if that is relevant). Given a (graded, finitely generated) module $M$ over $R$, is it possible to find a free module containing $M$? I see no reason for this to be true, and I can't find any references discussing it.

More generally, is there some class of rings for which (finitely presented) modules can be embedded into projective modules? Clearly this is true for self-injective rings, but I'm curious if it can hold for any rings which aren't self-injective.

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This isn't even true when $n=1$. Let $M=k[x]/(x)$. If you want to grade this put it all into the $0$-graded piece.

Embedding modules into projective modules amounts to embedding them in free modules. For a domain $R$ to embed $R/(a)$ in $R^n$ is doomed for nonzero $a$ as $R^n$ is torsion free.

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