The i-extended zero-divisor graphs of commutative rings

Abstract

The zero-divisor graphs of commutative rings have been used to build bridges between ring theory and graph theory. Namely, they have been used to characterize many ring properties in terms of graphic ones. However, many results are established only for reduced rings because a zero-divisor graph defined in the classical manner lacks the information on relationship between powers of zero-divisors. The aim of this article is to remedy this situation by introducing a parametrized family of graphs ${\{{\overline{\Gamma }}_i(R)\}}_{i\in {\mathbb{N}}^{*}},$ for a ring R, which reveals more of the relationship between powers of zero-divisors as follows: For each $i\in {\mathbb{N}}^{*},{\overline{\Gamma }}_i(R)$ is the simple graph whose vertex set is the set of non-zero zero-divisors such that two distinct vertices x and y are joined by an edge if there exist two positive integers $n\le i$ and $m\le i$ such that $x^n{y}^m=0$ with $x^n\ne 0$ and $y^m\ne 0.$ Our aim is to study in detail the behavior of the filtration ${\{{\overline{\Gamma }}_i(R)\}}_{i\in {\mathbb{N}}^{*}}$ as well as the relations between its terms. We give answers to several interesting and natural questions that arise in this context. In particular, we characterize girth and diameter of ${\overline{\Gamma }}_i(R)$ and give various examples.

Keywords:

• Extended zero-divisor graphs
• filtration of i-extended zero-divisor graphs
• zero-divisor graphs

2020 Mathematics Subject Classification:

• 13A15
• 13A99
• 13B99
• 13F99

Acknowledgments

Part of this work was done while Raja L’hamri visited the Institute of Mathematics of Jagiellonian University in Kraków (Poland) under Erasmus + KA107 scholarship grant. She would like to thank it for the warm hospitality as well as Professor Paweł Borówka for his kind support and encouragement during her visit. Also especial thanks go to Professor Bartłomiej Bosek, Professor Grzegorz Kapustka and all members of Institute of Mathematics and Institute of Computer Science.