Rings in which every element is the sum of a left zero-divisor and an idempotent

2019 ◽  
Vol 95 (3-4) ◽  
pp. 321-334
Author(s):  
Ebrahim Ghashghaei ◽  
Muhammet Tamer Kosan
Keyword(s):  
Zero Divisor ◽  
Left Zero

Commutativity of Near-rings with Derivations

2014 ◽  
Vol 21 (02) ◽  
pp. 215-230 ◽  
Author(s):  
Ahmed A. M. Kamal ◽  
Khalid H. Al-Shaalan
Keyword(s):  
Zero Divisor ◽  
Left Zero ◽  
Commutativity Theorems

In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d(a) is not a left zero-divisor in R for some a ∈ R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.

Rings in which every left zero-divisor is also a right zero-divisor and conversely

2019 ◽  
Vol 18 (05) ◽  
pp. 1950096 ◽  
Author(s):  
E. Ghashghaei ◽  
M. Tamer Koşan ◽  
M. Namdari ◽  
T. Yildirim
Keyword(s):  
Zero Divisor ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Natural Generalization ◽  
Left Zero ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

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.

On the left zero divisor graph of 3 × 3 matrices over ℤ2

2020 ◽  
Vol 41 (4) ◽  
pp. 1043-1060 ◽  
Author(s):  
Farkhanda Afzal ◽  
Faiza Khan Sherwani ◽  
Deeba Afzal ◽  
Faryal Chaudhry
Keyword(s):  
Zero Divisor ◽  
Left Zero ◽  
Zero Divisor Graph

On Domination in Zero-Divisor Graphs

2013 ◽  
Vol 56 (2) ◽  
pp. 407-411 ◽  
Author(s):  
Nader Jafari Rad ◽  
Sayyed Heidar Jafari ◽  
Doost Ali Mojdeh
Keyword(s):  
Zero Divisor ◽  
Domination Number ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Left Zero ◽  
Zero Divisor Graph

AbstractWe first determine the domination number for the zero-divisor graph of the product of two commutative rings with 1. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.

A Problem of Herstein on Group Rings

1974 ◽  
Vol 17 (2) ◽  
pp. 201-202 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Group Ring ◽  
Finite Subset ◽  
Zero Divisor ◽  
Formal Proof ◽  
Left Zero ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position ◽  
Finite Sum

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].

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 A CONDITION FOR SIMPLE RING IMPLYING FIELD II

1994 ◽  
Vol 25 (2) ◽  
pp. 163-166
Author(s):  
CHEN-TE YEN
Keyword(s):  
Zero Divisor ◽  
Left Zero ◽  
Simple Ring ◽  
Nonzero Idempotent

It is shown that if $R$ is a simple ring with identity 1 and with a nonzero idempotent $e$ and satisfies the condition $(P_2)_e$ : $(P_2)_e$    If $e- (a_1b_1+a_2b_2)$ is a right (left)zero divisor in $R$, then so is $e- (b_1a_1+b_2a_2)$. then $R$ is a field.Thus if $R$ is a simple ring then $eRe$ is a field for every nonzero idempotent $e$ in $R$ if it exists and $eRe$ satisfies $(P_2)_e$. We also discuss the above property for the simple ring case by eliminating the identity 1.

THE CYLINDRICAL CROSSING NUMBER OF ZERO DIVISOR GRAPHS

2020 ◽  
Vol 9 (8) ◽  
pp. 5901-5908
Author(s):  
M. Sagaya Nathan ◽  
J. Ravi Sankar
Keyword(s):  
Zero Divisor ◽  
Crossing Number
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