# tensor product of modules over commutative ring

It is well-know that the tensor product of two finitely generated modules over a given ring $R$ is finitely generated. Now my question is

Let $M$ and $N$ be $R$-modules where $R$ is a commutative ring. If $N$ is a FG module and we know that $M$ is not a FG module. Then what happens for the tensor product of $M$ and $N$ over $R$. Can we say that it is not FG or it may be FG. I am interested in the later case.

The tensor product may be finitely generated, even if both factors are not: For example consider the $\mathbf Z$-modules $M := \def\Z{\mathbf Z}\bigl(\Z/(3)\bigr)^{(\mathbf R)} = \bigoplus_{x\in \mathbf R} \Z/(3)$ and $N := \bigl(\Z/(2)\bigr)^{(\mathbf R)}$. Then neither $M$ nor $N$ is finitely generated, but $M \otimes N = 0$ is.
I do not exactly know, what you mean by "non-trivial", but the example can be modified, to have non-zero product, let $$M:= \bigl(\Z/(2)\bigr)^{(\Z)} \oplus \Z/(3), N := \Z(3)$$ then $M$ is not finitely generated, but $N$ is and $$M \otimes N = \Z/(3)$$ is also.