ZERO-DIVISOR GRAPH OF AN IDEAL OF A NEAR-RING

Abstract

In this paper, we associate the graph ΓI(N) to an ideal I of a near-ring N. We exhibit some properties and structure of ΓI(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor graph Γ(R) of R are equal. We prove that Beck's conjecture is true for ΓI(N). Moreover, we characterize all right permutable near-rings N for which the graph ΓI(N) is finitely colorable.