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.
Highlights
Throughout this paper all rings are commutative rings with unit element such that 1 ̸= 0
We denote the set of all zero-divisors, the set of all nilpotent elements, the set of all maximal ideals, the set of all minimal prime ideals, and the set of Jacobson radical of a ring R by Z(R), N il(R), M ax(R), M in(R) and J(R), respectively
Motivated by [17], we define the essential annihilating-ideal graph of R denoted by EG(R) with vertex set A∗(R) and two distinct vertices I1 and I2 adjacent if and only if Ann(I1I2) is an essential ideal of R
Summary
Throughout this paper all rings are commutative rings (not a field) with unit element such that 1 ̸= 0. In [11], Behboodi et al generalized the zero-divisor graph to ideals by defining the annihilating-ideal graph AG(R), with vertex set is A∗(R) and two distinct vertices I1 and I2 are adjacent if and only if I1I2 = 0. Nikmehr et al introduced the essential graph EG(R) with vertex set Z∗(R) = Z(R)\{0} and two distinct vertices x and y are adjacent if and only if annR(xy) is an essential ideal of R. Motivated by [17], we define the essential annihilating-ideal graph of R denoted by EG(R) with vertex set A∗(R) and two distinct vertices I1 and I2 adjacent if and only if Ann(I1I2) is an essential ideal of R. Rehman: On the essential annihilating-ideal graph of commutative rings 3 all the Artinian rings R for which EG(R) is isomorphic to some well-known graphs
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