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Can any module be included into a large enough free module? For example if I'm looking at some R-module M, then would $\underset{m \in M}{\bigoplus} R$ containt M as an R-module?

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No. Every Abelian Group is a $\mathbb Z$-module, but a free $\mathbb Z$-module is torsion free (has no elements of finite order). So this fails, for example, for any finite abelian group as a $\mathbb Z$-module.

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