If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero

Question

I am trying to work through the following problem:

If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero.

My thought on how to approach this problem is to suppose $M\otimes N=0$ and to use the universal property of the tensor product. Any biadditive map from $M\times N$ must then be the zero map by commutativity of the appropriate diagram.

I am unsure of what group I should map $M\times N$ biadditively to. Also I don't see where projectivity and finitely generated will come into play using this approach.

Perhaps there is a better way to approach the problem. Any hints pertaining to my approach or better ways to do the problem are welcome.

Answer 1

Jürgen's comment is right on the money: if $R = R_1 \times R_2$ then you can take $M = R_1, N = R_2$. These are both finitely generated and projective, but $M \otimes N = 0$.

You need an extra condition on either $M$ or $R$. I think a sufficient condition on $R$ is that it has no nontrivial idempotents, and a sufficient condition on $M$ is that it has full support in the sense that all of its localizations $M_p$ are nonzero.