# Answer to Question #16900 in Abstract Algebra for Irvin

Question #16900

Does this hold over an arbitrary ring R: any direct product of copies of V is semisimple?

Expert's answer

This does not hold (in general) without the commutativity assumption. To produce a counterexample, let

*V*be a right vector space over a division ring*D*with a basis*{e*1*, e*2*, . . . }*, and let*R*= End(*V*). Then_{D}*V*is a simple left*R*-module. The map*ϕ*:*:=*_{R}R −→ P*V × V ×· · ·*defined by*ϕ*(*r*) = (*re*1*, re*2*, . . .*) is easily checked to be an*R*-module isomorphism. In particular,*is cyclic. On the other hand,*_{R}P*is obviously not noetherian, so it is not semisimple.*_{R}P
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