Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring

Abstract

Let $ R, S $ be rings with unity, $ M $ a module over $ S $, where $ S $ a commutative ring, and $ f \colon R \rightarrow S $ a ring homomorphism. A ring representation of $ R $ on $ M $ via $ f $ is a ring homomorphism $ \mu \colon R \rightarrow End_S(M) $, where $ End_S(M) $ is a ring of all $ S $-module homomorphisms on $ M $. One of the important properties in representation of rings is the Schur's Lemma.  The main result of this paper is partly the generalization of Schur's Lemma in representations of rings on modules over a commutative ring