scholarly journals Rings with a few more zero-divisors

1971 ◽  
Vol 5 (2) ◽  
pp. 271-274 ◽  
Author(s):  
C. Christensen
Keyword(s):  
Zero Divisor ◽  
Finite Ring ◽  
Left Zero ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unital Rings

It is well-known that every finite ring with non-zero-divisors has order not exceeding the square of the order n of its left zero-divisor set. Unital rings whose order is precisely n2 have been described already. Here we discuss finite rings with relatively larger zero-divisor sets, namely those of order greater than n3/2. This is achieved by describing the class of all finite rings with left composition length two at most, and using a theorem relating the left composition length of a finite ring to the size of its left zero-divisor set.

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.

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.

scholarly journals On the Zero Divisor Graphs of a Class of Commutative Completely Primary Finite Rings

2016 ◽  
Vol 12 (3) ◽  
pp. 6021-6026
Author(s):  
Maurice Oduor ◽  
Walwenda Shadrack Adero
Keyword(s):  
Maximal Ideal ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

Let R be a Completely Primary Finite Ring with a unique maximal ideal Z(R)), satisfying ((Z(R))n−1 ̸= (0) and (Z(R))n = (0): The structures of the units some classes of such rings have been determined. In this paper, we investigate the structures of the zero divisors of R:

On the Set of Zero Divisors of a Topological Ring

1967 ◽  
Vol 10 (4) ◽  
pp. 595-596 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Finite Number ◽  
Compact Subset ◽  
Finite Ring ◽  
Left Zero ◽  
Topological Ring ◽  
Compact Ring ◽  
Zero Divisors ◽  
Interesting Corollary

Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra are closed subsets of R. We will prove that if the set of non - trivial right (left) zero divisors of R is a non-empty set and the set of all right (left) zero divisors of R is a compact subset of R, then R is a compact ring. This theorem has an interesting corollary. Namely, if R is a discrete ring with a finite number of non - trivial left or right zero divisors then R is a finite ring (Refer [1]).

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 Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

2021 ◽  
pp. 137-154
Author(s):  
Hezron Saka Were ◽  
Maurice Oduor Owino ◽  
Moses Ndiritu Gichuki
Keyword(s):  
Maximal Ideal ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

scholarly journals Automorphisms of Zero Divisor Graphs of Cube Radical Zero Completely Primary Finite Rings

2020 ◽  
pp. 83-90
Author(s):  
Lao Hussein Mude ◽  
Owino Maurice Oduor ◽  
Ojiema Michael Onyango
Keyword(s):  
Algebraic Structure ◽  
Zero Divisor ◽  
Finite Rings ◽  
Zero Divisors

One of the most interesting areas of research that has attracted the attention of many scholars are theory of zero divisor graphs. Most recent research have focused on properties of zero divisor graphs with little attention given on the automorphsisms, despite the fact that automorphisms are useful in interpreting the symmetries of algebraic structure. Let R be a commutative unital finite rings and Z(R) be its set of zero divisors. In this study, the automorphisms zero divisor graphs of such rings in which the product of any three zero divisor is zero has been determined.

The maximal degree of a zero-divisor graph

2019 ◽  
pp. 2050151
Author(s):  
Husam Q. Mohammad ◽  
Nazar H. Shuker ◽  
Luma A. Khaleel
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finite Ring ◽  
Maximal Degree ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Finite Commutative Rings

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

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.

The zero-divisor graph of a ring with involution

2018 ◽  
Vol 17 (03) ◽  
pp. 1850050 ◽  
Author(s):  
Avinash Patil ◽  
B. N. Waphare
Keyword(s):  
Complete Graph ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Sufficient Conditions ◽  
Left Zero ◽  
Star Graph ◽  
Artinian Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

For a *-ring [Formula: see text], we associate a simple undirected graph [Formula: see text] having all nonzero left zero-divisors of [Formula: see text] as vertices and, two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having [Formula: see text] a complete graph or a star graph, and sufficient conditions are obtained for [Formula: see text] to be connected and also for [Formula: see text] to be disconnected. For a Rickart *-ring [Formula: see text], we characterize the girth of [Formula: see text] and prove a sort of Beck’s conjecture.

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