Geodesic
Extended Annihilating-Ideal Graph of a Commutative Ring
Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 2, pp. 279-291

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Let R be a commutative ring with identity. An ideal I of a ring R is called an annihilating-ideal if there exists a nonzero ideal J of R such that IJ = (0) and we use the notation 𝔸(R) for the set of all annihilating-ideals of R. In this paper, we introduce the extended annihilating-ideal graph of R, denoted by 𝔼𝔸𝔾(R). It is the simple graph with vertices 𝔸(R)^∗ = 𝔸(R) {(0)}, and two distinct vertices I and J are adjacent whenever there exist two positive integers n and m such that I^nJ^m = (0) with I^n ≠ (0) and J^m ≠ (0). Here we discuss in detail the diameter and girth of 𝔼𝔸𝔾(R) and investigate the coincidence of 𝔼𝔸𝔾(R) with the annihilating-ideal graph 𝔸𝔾(R). Moreover we propose open questions in this paper.
Keywords: annihilating-ideal graph, extended annihilating-ideal graph 
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Nithya, S.; Elavarasi, G. Extended Annihilating-Ideal Graph of a Commutative Ring. Discussiones Mathematicae. General Algebra and Applications, Tome 42 (2022) no. 2, pp. 279-291. http://geodesic.mathdoc.fr/item/DMGAA_2022_42_2_a2/