Visweswaran, S., Vadhel, P. (2018). Some results on a subgraph of the intersection graph of ideals of a commutative ring. Journal of Algebra and Related Topics, 6(2), 35-61. doi: 10.22124/jart.2018.11188.1114

S. Visweswaran; P. Vadhel. "Some results on a subgraph of the intersection graph of ideals of a commutative ring". Journal of Algebra and Related Topics, 6, 2, 2018, 35-61. doi: 10.22124/jart.2018.11188.1114

Visweswaran, S., Vadhel, P. (2018). 'Some results on a subgraph of the intersection graph of ideals of a commutative ring', Journal of Algebra and Related Topics, 6(2), pp. 35-61. doi: 10.22124/jart.2018.11188.1114

Visweswaran, S., Vadhel, P. Some results on a subgraph of the intersection graph of ideals of a commutative ring. Journal of Algebra and Related Topics, 2018; 6(2): 35-61. doi: 10.22124/jart.2018.11188.1114

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

^{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 $R$ by $\mathbb{I}(R)$ and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $\mathbb{I}(R)^{*}$ 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 $R$.