On the essential annihilating-ideal graph of commutative rings

Mohd Nazim, Nadeem ur Rehman

Abstract


Let R be a commutative ring with unity, A(R) be the set of all annihilating-ideals of R and A*(R) = A(R) \ {0}. In this paper, we introduced and studied the essential annihilating-ideal graph of R, denoted by EG(R), with vertex set A*(R) and two distinct vertices I1 and I2 are adjacent if and only if Ann(I1I2) is an essential ideal of R. We prove that EG(R) is a connected graph with diameter at most three and girth at most four if EG(R) contains a cycle. Furthermore, the rings R are characterized for which EG(R) is a star or a complete graph. Finally, we classify all the Artinian rings R for which EG(R) is isomorphic to some well-known graphs.


Keywords


Annihilating-ideal graph, zero-divisor graph, complete graph, planar graph, genus of a graph

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DOI: https://doi.org/10.26493/1855-3974.2645.8fc

ISSN: 1855-3974

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