Compressed intersection annihilator graph

Mayssa Soliman ,Nefertiti Megahed

doi:10.28919/jmcs/4787

Abstract

Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by $IA(R)$, and investigate some of its theoretical properties and its relation with the structure of the ring. It is a generalization of the torsion graph $\Gamma_{R}(R)$. We study classes of rings for which the equivalence between the set of zero-divisors of $R$ being an ideal and the completeness of $IA(R)$ holds. We also study the relation between $\Gamma_{R}(R)$ and $IA(R)$. In addition, we show that if the compressed intersection annihilator graph of a ring $R$ is finite, then there exists a subring $S$ of $R$ such that $IA(S)\cong IA(R)$. Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph $IA(R)$ with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when $R$ is the ring of integers modulo $n$, the direct product of integral domains, the direct product of Artinine local rings and the direct product of two rings such that one of them is not an integral domain.

Highlights

The study of zero-divisors plays an important role in the ring theory, for example to find solutions to equations, the set of zero-divisors lacks algebraic structure
Beck in 1988, [10], when he defined the zero-divisor graph Γ(R) as the undirected simple graph with vertices represented by all elements of R and two distinct vertices are adjacent if their product is zero
The set of vertices was constructed from equivalence classes of zero-divisors determined by the following equivalence relation ∼ on R: x ∼ y if and only if annR(x) = annR(y), for any x, y ∈ R, where annR(x) = {r ∈ R | rx = 0}

Summary

Introduction

The study of zero-divisors plays an important role in the ring theory, for example to find solutions to equations, the set of zero-divisors lacks algebraic structure. Badawi in 2008, [3], is the total graph of a commutative ring denoted by T (Γ(R)) It is defined as the undirected simple graph with R as the set of vertices and two distinct vertices are adjacent if and only if their sum is a zero-divisor. The set of vertices of ΓR(M) is the set of non-zero torsion elements T (M)∗ where, T (M)∗ = {m ∈ M | annR(m) = {0}} with two distinct vertices x and y are adjacent if and only if annR(x) ∩ annR(y) = {0} In their work, they studied in which case ΓR(M) is connected with diam(ΓR(M)) is less than or equal to three, the relationship between the diameter of ΓR(M) and ΓR(R) and proved that the girth of ΓR(M) belongs to {3, ∞}. R is said to have a finite Goldie dimension if it does not contain infinite direct sums of non-zero ideals

Compressed Intersection Annihilator Graph

Diameter and Girth

Compressed Intersection Annihilator Graph of Zn

Compressed Intersection Annihilator Graph of a Finite Product of Rings