Commuting graphs of some subsets in simple rings

Abstract

Let D be a division ring with center F and n ⩾ 1 a natural number. For S ⊆ M n ( D) the commuting graph of S, denoted by Γ( S), is the graph with vertex set S⧹ Z( S) such that distinct vertices a and b are adjacent if and only if ab = ba. In this paper we prove that if n > 2 and A , N , I , T are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, Γ ( A ) , Γ ( N ) , Γ ( I ) , and Γ ( T ) are connected graphs. We show that if n > 2 and U is the set of all upper triangular matrices, then for every algebraic division ring D, Γ ( U ) is a connected graph. Also it is shown that if R is the set of all reducible matrices and M n ( D) is algebraic over F, then for n > 2, Γ ( R ) is a connected graph. Finally, we prove that for n ⩾ 2, Γ ( M n ( H ) ) is a connected graph, where H is the ring of real quaternions.