On commuting graphs of semisimple rings

Abstract

Let R be a non-commutative ring. The commuting graph of R denoted by Λ (R), is a graph with vertex set R \ Z(R), and two distinct vertices a and b are adjacent if ab = ba. In this paper we investigate some properties of Λ(R), whenever R is a finite semisimple ring. For any finite field F, we obtain minimum degree, maximum degree and clique number of Λ(M n (F)). Also it is shown that for any two finite semisimple rings R and S, if Λ(R) ≃ Λ(S), then there are commutative semisimple rings R1 and S1 and semisimple ring T such that R ≃T × R1, S ≃ T × S1 and |R1| = |S1|. © 2004 Elsevier Inc. All rights reserved.