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.

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.

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.

scholarly journals THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY

2020 ◽  
Vol 83 (1) ◽  
pp. 127-132
Author(s):  
Nurhidayah Zaid ◽  
Nor Haniza Sarmin ◽  
Sanhan Muhammad Salih Khasraw
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Random Elements

Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.

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.

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

Associate class graph of zero-divisors of a commutative ring

2019 ◽  
Vol 19 (08) ◽  
pp. 2050155
Author(s):  
Gaohua Tang ◽  
Guangke Lin ◽  
Yansheng Wu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
New Class ◽  
Associate Class ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Class Graph

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

scholarly journals When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Zero Divisors

Let be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of has at least one edge. Under the above assumptions on , it is shown in this paper that the complement of the zero-divisor graph of is complemented if and only if is isomorphic to as rings. Moreover, if is not isomorphic to as rings, then, it is shown that in the complement of the zero-divisor graph of , either no vertex admits a complement or there are exactly two vertices which admit a complement.

Characterizations of Three Classes of Zero-Divisor Graphs

2012 ◽  
Vol 55 (1) ◽  
pp. 127-137 ◽  
Author(s):  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Central Vertex ◽  
Integral Domains ◽  
Boolean Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

Sign in / Sign up

Export Citation Format

Share Document