On zero-divisor graphs of finite rings

  title={On zero-divisor graphs of finite rings},
  author={Saieed Akbari and Ali Mohammadian},
  journal={Journal of Algebra},

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is

Construction of Zero Divisor Graphs of Rings

If R is a commutative ring, Z(R) is the set of zero-divisor of R and Z(R) = Z(R) − {0}, then the zero-divisor graph of R, Γ (Z*(R)) usually written as Γ(R), is the graph in which each element of

A Note on the Uniqueness of Zero-Divisor Graphs with Loops (Research)

It is known that rings which have isomorphic zero-divisor graphs are not necessarily isomorphic. Zero-divisor graphs for rings were originally defined without loops because edges are only defined on

Metric and upper dimension of zero divisor graphs associated to commutative rings

Abstract Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices

On the metric dimension of a zero-divisor graph

ABSTRACT Let R be a commutative ring with unity 1 and let G(V,E) be a simple graph. In this research article, we study the metric dimension in zero-divisor graphs associated with commutative rings.


The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge

On the nilpotent graph of a ring

Let R be a ring with unity. The nilpotent graph of R , denoted by ΓN (R) , is a graph with vertex set ZN (R) ∗ = {0 � x ∈ R | xy ∈ N (R) for some 0 � y ∈ R} ; and two distinct vertices x and y are


Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two

On zero-divisor graphs of Boolean rings

The zero-divisor graph of a ring R is the graph whose vertices consist of the nonzero zero-divisors of R in which two distinct vertices a and b are adjacent if and only if either ab = 0 or ba = 0. In

Eulerian Zero-Divisor Graphs

AbstractIn this article, we characterize for which nite commutative ringR, the zero-divisor graph ( R), the line graph L(( R)), the com-plement graph ( R), and the line graph for the complement



Zero-divisor graphs of non-commutative rings

Structure in the Zero-Divisor Graph of a Non-Commutative Ring

In a manner analogous to the commutative case, the zero-divisor graph of a noncommutative ring R can be defined as the directed graph G. It has been shown that G is not a tournament if R is a finite

The annihilator of radical powers in the modular group ring of a $p$-group

We show that if N is the radical of the group ring and L is the exponent of N, then the annihilator of NW is NL-W+1. As corollaries we show that the group ring has exactly one ideal of dimension one

The Zero-Divisor Graph of a Commutative Ring☆

For each commutative ring R we associate a (simple) graph Γ(R). We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R).

A first course in noncommutative rings

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the

A Simple Proof of a Theorem of Schur

I. J. A. Gallian and J. Van Buskirk, The number of homomorphisms from Zm into Z,, Amer. Math. Momhly 91 (1984) 196-197. 2. J. A. Gallian and D. S. Jungreis, Homomorphisms from Zm[i) into Z,[i) and

The algebraic structure of group rings

$m_{i}$ . Then $A_{i}$ has the rank $r_{i}q_{i}^{2}m_{i}^{2}$ over K. We shall call the numbers $n\ell_{i}$ the Sckur indice $s^{\neg}$ of $\mathfrak{G}$ , since they first occurred in the work of 1.