Question

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

Answer 1

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(V_D). 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.