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.

摘要

交换环的零因数图已被用于在环论和图论之间架起桥梁。也就是说，它们已被用于根据图形特性来表征许多环特性。然而，许多结果仅针对缩减环建立，因为以经典方式定义的零因数图缺乏有关零因数的幂之间关系的信息。本文的目的是通过引入参数化图族来纠正这种情况${\{{\overline{\Gamma }}_{\mathrm{一世}}(\mathrm{电阻})\}}_{\mathrm{一世}\in {\mathbb{N}}^{*}},$对于环R，它揭示了更多零因数的幂之间的关系如下：对于每个$\mathrm{一世}\in {\mathbb{N}}^{*},{\overline{\Gamma }}_{\mathrm{一世}}(\mathrm{电阻})$是一个简单的图，其顶点集是一组非零零因数，如果存在两个正整数，则两个不同的顶点x和y由一条边连接$n\le \mathrm{一世}$ 和 $米\le \mathrm{一世}$ 以至于 $X^n{是}^{米}=0$ 和 $X^n\ne 0$ 和 ${是}^{米}\ne 0.$ 我们的目标是详细研究过滤的行为 ${\{{\overline{\Gamma }}_{\mathrm{一世}}(\mathrm{电阻})\}}_{\mathrm{一世}\in {\mathbb{N}}^{*}}$以及其术语之间的关系。我们回答了在这种情况下出现的几个有趣且自然的问题。特别是，我们描述了周长和直径的特征${\overline{\Gamma }}_{\mathrm{一世}}(\mathrm{电阻})$ 并举出各种例子。