Some results on a subgraph of the intersection graph of ideals of a commutative ring

Document Type : Research Paper

Authors

¹ Department of Mathematics, Saurashtra University, Rajkot, India.

² Department of Mathematics, Saurashtra University, Rajkot, India

Abstract

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let

R

be a ring. Let us denote the collection of all proper ideals of

R

I (R)

and

I (R) ∖ {(0)}

I (R)^{*}

. With

R

, we associate an undirected graph denoted by

g (R)

, whose vertex set is

I (R)^{*}

and distinct vertices

I_{1}, I_{2}

are adjacent in

g (R)

if and only if

I_{1} \cap I_{2} \neq I_{1} I_{2}

. The aim of this article is to study the interplay between the graph-theoretic properties of

g (R)

and the ring-theoretic properties of

R

Keywords