A graph which recognizes idempotents of a commutative ring

TL;DR

This paper introduces a graph based on ideals of a commutative ring, where adjacency reflects the product and intersection of ideals, revealing insights into the ring's structure.

Contribution

It defines a novel graph on ring ideals and explores its properties and connections to the algebraic structure of the ring.

Findings

The graph's properties relate to the ring's ideal structure.

Characterization of idempotent ideals via the graph.

Insights into the ring's decomposition from the graph analysis.

Abstract

In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=I\cap J$. We obtain some properties of this graph and study its relation to the structure of $R$.