Question

Let M be a finitely generated left R-module and E = End(RM). Show that if R is semisimple, then so is E.

Answer 1

First assume R is semisimple,and let S1, . . . , Sr be a complete set of simple left R-modules.Then M = M1 ⊕· · ·⊕Mr, where Mi is the sum of allsubmodules of M which are isomorphic to Si. Since M isfinitely generated, Mi ∼ niSi for suitable integers ni.Therefore, EndRM ∼ (product on i) EndRMi ∼(product on i) Mni(EndRSi).By Schur’s Lemma, all EndRSi are division rings, so EndRM issemisimple.