Question

Let U, V be modules over a commutative ring R. If U or V is semisimple, show that U ⊗R V is also semisimple.

Answer 1

For the first statement, it suffices to handle the case where V is semisimple. Since tensor product distributes over direct sums, we may further assume that V is simple, say V ∼ R/m where m is a maximal ideal of R. But then m · (U ⊗_R V ) = 0, so U ⊗_R V is a vector space over the field R/m. Thus, U ⊗_R V is semisimple over R/m, and also over R.