Let $k$ denote an algebraically closed field of characteristic $0$ (maybe characteristic is not important here, but certainly algebraic closure is). Let $A$ and $B$ denote finite-dimensional associative algebras over $k$. Let $M$ be an indecomposable $A$-module and $N$ an indecomposable $B$-module (both $M$ and $N$ finite-dimensional, of course).
Then is the external tensor product module $M\boxtimes_k N$ indecomposable over $A\otimes_k B$?
