Abstract. Let I be a proper ideal of a commutative ring R with 1 = 0. The ideal-based zero-divisor graph of R with respect to I, denoted by ΓI (R), is the (simple) graph with vertices { x ∈ R \ I | xy ∈ I for some y ∈ R \ I }, and distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, we study ΓI (R) for commutative rings R such that R/I is a chained ring.

Let R be a semiring with identity. In this paper, we introduce the intersection graph of co-ideals, denoted by G(R). The vertices of G(R) are non-trivial co-ideals of R, and two distinct vertices I and J are adjacent if and only if I \ J 6 = f1g. The basic properties and possible structures of this graph are studied and the interplay between the algebraic properties of R and the graph-theoretic structure of G(R) is investigated.

Let R be a commutative ring with identity and let M be a prime Rmodule. Let R(+)M be the idealization of the ring R by the R-module M . We study the diameter and girth of the zero-divisor graph of the ring R(+)M . Mathematics Subject Classification: 13A99, 13A15

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.