Question: R is an integral domain, M is a projective R-module and Tor(M) = {$x\in M$: rx = 0 for some non-zero $r\in R$}. Prove that M is torsion-free (i.e. Tor(M) = {$0_M$}).
Attempt: I don't seem to know how to tackle this problem. I thought of a R-module homomorphism and surjective function $f: N\to M$. Since M is projective, then there exists the inverse $g:M\to N$. however, I couldn't justify introducing N, and I couldn't make progress as well.
I would really appreciate your help. Thanks.
The map $\bigoplus_{x\in M} R \to M$ is surjective, hence there exists a section map going to other way which exhibits $M$ as a submodule of a free module. Since $R$ is an integral domain, any $R$-submodule of a free $R$-module is torsion-free.