Compressed Intersection Annihilator Graph

Mayssa Soliman; Nefertiti Megahed

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arXiv · 2002.05450

Compressed Intersection Annihilator Graph

Mayssa Soliman·Nefertiti Megahed

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Abstract

Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by $IA(R)$, and investigate some of its theoretical properties and its relation with the structure of the ring. It is a generalization of the torsion graph $\Gamma_{R}(R)$. We study classes of rings for which the equivalence between the set of zero-divisors of $R$ being an ideal and the completeness of $IA(R)$ holds. We also study the relation between $\Gamma_{R}(R)$ and $IA(R)$. In addition, we show that if the compressed intersection annihilator graph of a ring $R$ is finite, then there exists a subring $S$ of $R$ such that $IA(S)\cong IA(R)$. Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph $IA(R)$ with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when $R$ is the ring of integers modulo $n$, the direct product of integral domains, the direct product of Artinine local rings and the direct product of two rings such that one of them is not an integral domain.

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Mayssa Soliman, Nefertiti Megahed. 2020-02-13. Compressed Intersection Annihilator Graph. https://arxiv.org/abs/2002.05450

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