Commuting graphs of full matrix rings over finite fields

Abstract

Let R be a non-commutative ring and Z (R) be its center. The commuting graph of R is defined to be the graph Γ (R) whose vertex set is R {minus 45 degree rule} Z (R) and two distinct vertices are joint by an edge whenever they commute. Let F be a finite field, n ≥ 2 an arbitrary integer and R be a ring with identity such that Γ (R) ≅ Γ (Mn (F)), where Mn (F) is the ring of n × n matrices over F. Here we prove that | R | = | Mn (F) |. We also show that if | F | is prime and n = 2, then R ≅ M2 (F). © 2008 Elsevier Inc. All rights reserved.