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*_{D}). Then *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, _{R}P is cyclic. On the other hand, _{R}P is obviously not noetherian, so it is not semisimple.

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