On projective intersection graph of ideals of commutative rings

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

Intersection graphs of quasinormal subgroups of general skew linear groups

“…Recently, the intersection graphs whose vertices are sub-structures of groups, rings, or modules are studied deeply in the works of Akbari et al (see [1][2][3]). More dated results can be found in [4,15].…”

A study of upper ideal relation graphs of rings

“…Pucanović et al [25] characterized all the graph classes of genus 2 that are intersection graphs of ideals of some commutative rings. The intersection ideal graphs with crosscap at most two of all the Artinian commutative rings have been investigated by Ramanathan [26]. To reveal the relationships between the ideals and the elements of the ring R, contained in the ideal (z).…”