Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero. Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero.
I know one way to show this is to find an R-bilinear map which is nonzero, but am not sure how to find it.

This is false. Let $R=\mathbb{C}[X]/(X^2)$. Let $x=\bar{X}$, so $x^2=0$, and let $M= R x$. Then $M$ is a finitely generated nonzero $R$-module, but $M\otimes M$ is generated by $x\otimes x=x^2\otimes 1=0\otimes 1=0$.
More generally, if $R$ is a commutative ring having a nonzero ideal $I$ satisfying $I^2=0$, then $I\otimes_R I=0$ and you get a counterexample.
