I'm a bit embarrassed to ask this, but I've gotten myself confused over what I think is a simple issue. Let $A$ be a local ring, $k$ its residue field, and $M,N$ finitely generated $A$-modules. An exercise in Atiyah and Macdonald asserts that if $M \otimes_A N = 0$, then either $M = 0$ or $N = 0$. They give a hint in which they use the notation $M_k = M \otimes_A k$: it says $M \otimes_A N = 0$ implies $(M \otimes_A N)_k = 0$ implies $M_k \otimes_k N_k = 0$. I don't fully understand what happened in the second step.
This motivates the following more general question: if $A$ is any commutative ring and $B$ a commutative $A$-algebra, and $M,N$ are $B$-modules, can we identify $M \otimes_A N \cong M \otimes_B N$ as, say, $A$-modules? I think this is either very obvious or very naive.