Finite tensor product of infinite $R$-modules

Question

I was challenged to find a ring $R$ and infinite $R$-modules $A$ and $B$ such that $A \otimes_R B$ is a finite but nontrivial abelian group.

I can think of many examples that give $0$, but I'm stumped when it comes to nontrivial finite examples. I'm used to working mostly with $\mathbb{Z}$-modules but I haven't gotten anywhere with those yet. So perhaps a more clever choice of $R$ is all I need.

Answer 1

Let $R = \mathbb F_p[x, y]$ and let $A$ and $B$ be $R/x$ and $R/y$ respectively.