ON THE IDEMPOTENT GRAPH OF A RING

Abstract

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper we show if D is a division ring, then the clique number of I(Mn(D))(n ≥ 2) is n and for any commutative Artinian ring R the clique number and the chromatic number of I(R) are equal to the number of maximal ideals of R. We prove that for every left Noetherian ring R, the clique number of I(R) is finite. For every finite field F, we also determine an independent set of I(Mn(F)) with maximum size. If F is an infinite field, then we prove that the domination number of I(Mn(F)) is infinite. We show that the idempotent graph of every reduced ring is connected and if n ≥ 3 and D is a division ring, then I(Mn(D)) is connected and moreover diam(I(Mn(D))) ≤ 5.