$A \otimes_k B$-module $M \otimes_k N$ simple?

Question

Let $A$ and $B$ be nice $k$-algebras. If I have simple modules $M$ and $N$, over $A$ and $B$ respectively, is the $A \otimes_k B$-module $M \otimes_k N$ simple?

Answer 1

I'm not sure what you count as "nice", but here's a counterexample. Let $k=\mathbb{R}$ and $A=B=M=N=\mathbb{C}$. Then $A\otimes_k B=\mathbb{C}\otimes_\mathbb{R}\mathbb{C}$ is not simple as a module over itself, being the direct sum of the submodules generated by $i\otimes 1-1\otimes i$ and $i\otimes 1+1\otimes i$. More generally, if $A=B$ is a field extending $k$, then $A\otimes_k B$ will never be a field (unless $A=B=k$).