Every projective module is a submodule of a free module?

Question

I've seen this statement on the internet but I could not find a proof. Actually this is true for any module I think. Can a proof be given as follows?

Let $M$ be an $R$-module. Take a generating set $X$ of $M$ over $R$. Then consider the free $R$-module $F$ over the set $X$. Hence $M \subset F$ and we are done. I hope this is not a nonsense idea.

Every projective module is isomorphic to a direct factor of a free $R$-module. — Bernard, Jan 20 '19 at 13:53

score 2 · Answer 1 · answered Jan 20 '19 at 13:52

2

What you get with your $X$ (a generating set of $M$) and your $F$ (free on $X$) is a surjective homomorphism $\pi:F\to M$, not an injection $M\to F$.

But if $M$ is projective, the surjection $\pi$ splits, that is there's an injective homomorphism $\iota:M\to F$ with $\pi\circ\iota=\text{id}_M$.

answered Jan 20 '19 at 13:52

Angina Seng

158,341

Ok, that's the importance of projectivity! Thanks man. – Philip Johnson Jan 20 '19 at 13:59

Every projective module is a submodule of a free module?

1 Answers1