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

Source abstract

In this paper, we associate the graph Γ I (N) to an ideal I of a near-ring N. We exhibit some properties and structure of Γ I (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 Γ I (N). Moreover, we characterize all right permutable near-rings N for which the graph Γ I (N) is finitely colorable.