Some graph on prime ideals of a commutative ring

Year 2018, Volume: 23 Issue: 23, 157 - 166, 11.01.2018

Alpesh M. Dhorajia

https://doi.org/10.24330/ieja.373659

Cited By: 1

Abstract

Let R be a commutative ring with an identity. Let Spec(R) be

the set of all prime ideals of R and Max(R) be the set of all maximal ideals

of R. Let S  Max(R). We dene an S-proper ideal sum graph on Spec(R),

denoted by 􀀀S(Spec(R); S), as an undirected graph whose vertex set is the set

Spec(R) and, for two distinct vertices P and Q, there is an arc from P to Q,

whenever P +Q M, for some maximal idealMin S. In this paper, we prove

that the complement graph of a proper sum graph 􀀀(Spec(R); S) is complete

if and only if R is an Artinian ring. We also study some basic properties of

the graph 􀀀S(Spec(R); S) such as connectivity, girth and clique number. We

explore the in

uence of the ring theoretic properties of a commutative ring R

on the proper sum graph of R and vice versa.

Keywords

Proper sum graph, Artinian ring, commutative ring

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