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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 {e1, e2, . . . }, and let R = End(VD). Then V is a simple left R-module. The map ϕ: RR −→ P := V × V ×· · · defined by ϕ(r) = (re1, re2, . . . ) is easily checked to be an R-module isomorphism. In particular, RP is cyclic. On the other hand, RP is obviously not noetherian, so it is not semisimple.

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