# Answer to Question #16897 in Abstract Algebra for Irvin

Question #16897

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

Expert's answer

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*.
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