Idempotent Divisor Graph of Commutative Ring

Husam Q Mohammad,Nazar H Shuker

doi:10.24996/ijs.2022.63.2.21

Abstract

This work aims to introduce and to study a new kind of divisor graph which is called idempotent divisor graph, and it is denoted by . Two non-zero distinct vertices v1 and v2 are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of , as well as it’s connection with . We also study planarity of this graph.

Highlights

Let be a finite commutative ring with unity
We study planarity of this graph
In [1], Beck introduced the idea that connects between ring theory and graph theory when studied the coloring of commutative ring

Summary

Introduction

Let be a finite commutative ring with unity. We denote, and the set of zero divisors, the set of idempotent elements and the set of unit elements respectively.In [1], Beck introduced the idea that connects between ring theory and graph theory when studied the coloring of commutative ring. A graph G has radius 1 if and only if G contains a vertex u adjacent to all other vertices of G. A ring R is called Boolean, if every element is an idempotent. 2- If has only idempotent elements 0 and 1, local , 3- If finite non local ring, has idempotent element distinct .

Objectives

Results

Conclusion