On zero-divisor graphs of skew polynomial rings over non-commutative rings

2017 ◽  
Vol 16 (03) ◽  
pp. 1750056 ◽  
Author(s):  
E. Hashemi ◽  
R. Amirjan ◽  
A. Alhevaz
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Polynomial Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Skew Polynomial Rings ◽  
Derivation Formula ◽  
Base Ring ◽  
Skew Polynomial ◽  
Associative Rings

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950126 ◽  
Author(s):  
Ebrahim Hashemi ◽  
Mona Abdi ◽  
Abdollah Alhevaz
Keyword(s):  
Zero Divisor ◽  
Undirected Graph ◽  
Complete Characterization ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Noncommutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Base Ring

Let [Formula: see text] be an associative ring with nonzero identity. The zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonzero zero-divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Base Ring ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

FINITE RINGS WITH EULERIAN ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250055 ◽  
Author(s):  
A. S. KUZMINA
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Noncommutative Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

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 if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

The McCoy condition on skew monoid rings

2017 ◽  
Vol 10 (03) ◽  
pp. 1750050 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Polynomial Rings ◽  
Monoid Ring ◽  
Zero Divisor Graph ◽  
Monoid Homomorphism ◽  
Type Property

Let [Formula: see text] be an associative ring with identity, [Formula: see text] a monoid and [Formula: see text] a monoid homomorphism. When [Formula: see text] is a u.p.-monoid and [Formula: see text] is a reversible [Formula: see text]-compatible ring, then we observe that [Formula: see text] satisfies a McCoy-type property, in the context of skew monoid ring [Formula: see text]. We introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible [Formula: see text]-compatible rings and also various examples of [Formula: see text]-McCoy rings are provided. As an application of [Formula: see text]-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text].

scholarly journals ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250019 ◽  
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

NILPOTENT FINITE RINGS WITH PLANAR ZERO-DIVISOR GRAPHS

2008 ◽  
Vol 01 (04) ◽  
pp. 565-574 ◽  
Author(s):  
A. S. KUZ'MINA ◽  
Yu. N. MALTSEV
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph with all vertices non-zero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge iff xy = 0 or yx = 0 ([10]). In the present paper, we describe all nilpotent finite rings with planar zero-divisor graphs.

scholarly journals On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hafiz Muahmmad Afzal Siddiqui ◽  
Ammar Mujahid ◽  
Muhammad Ahsan Binyamin ◽  
Muhammad Faisal Nadeem
Keyword(s):  
Zero Divisor ◽  
Research Work ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Metric Dimension ◽  
Gaussian Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Algebraic Properties ◽  
Do So

Given a finite commutative unital ring S having some non-zero elements x ,   y such that x . y = 0 , the elements of S that possess such property are called the zero divisors, denoted by Z S . We can associate a graph to S with the help of zero-divisor set Z S , denoted by ζ S (called the zero-divisor graph), to study the algebraic properties of the ring S . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S . To do so, we will discuss the zero-divisor graphs for the ring of integers ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ S . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.

Undirected Zero-Divisor Graphs and Unique Product Monoid Rings

2019 ◽  
Vol 26 (04) ◽  
pp. 665-676
Author(s):  
Ebrahim Hashemi ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Shortest Path ◽  
Associative Ring ◽  
Zero Divisor ◽  
Unique Product ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Unique Product Monoid

Let R be an associative ring with identity and Z*(R) be its set of non-zero zero-divisors. The undirected zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and where two distinct vertices r and s are adjacent if and only if rs = 0 or sr = 0. The distance between vertices a and b is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the superimum of these distances. In this paper, first we prove some results about Γ(R) of a semi-commutative ring R. Then, for a reversible ring R and a unique product monoid M, we prove 0≤ diam(Γ(R))≤ diam(Γ(R[M]))≤3. We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])), strictly in terms of the properties of a ring R, where R is a reversible ring and M is a unique product monoid. Moreover, an example showing the necessity of our assumptions is provided.

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