Coloring in essential annihilating-ideal graphs of commutative rings

The essential annihilating-ideal graph epsilon G(R) of a commutative unital ring R is a simple graph, whose vertices are non-zero ideals of R with non-zero annihilator and there exists an edge between two distinct vertices I, J if and only if Ann(I J) has a non-zero intersection with any non-zero ideal of R. In this paper, we show that epsilon G(R) is weakly perfect, if R is Noetherian and an explicit formula for the clique number of epsilon G(R) is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number 2 are fully determined. Among other results, twin-free clique number and edge chromatic number of epsilon G(R) are examined.

Automated summary

This paper investigates the properties of the essential annihilating-ideal graph of a ring, demonstrates that under certain conditions this graph is weakly perfect, and provides an explicit formula to calculate its clique number. Furthermore, it fully determines the structures of rings whose essential annihilating-ideal graphs have chromatic number 2, and examines properties such as twin-free clique number and edge chromatic number.