Commuting graphs of some subsets in simple rings

Let D be a division ring with center F and n 1 a natural number. For S Mn(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 are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, , , , and are connected graphs. We show that if n > 2 and is the set of all upper triangular matrices, then for every algebraic division ring D, is a connected graph. Also it is shown that if is the set of all reducible matrices and Mn(D) is algebraic over F, then for n > 2, is a connected graph. Finally, we prove that for n 2, is a connected graph, where is the ring of real quaternions.