scholarly journals A Commutativity Theorem for Near-Rings

1977 ◽  
Vol 20 (1) ◽  
pp. 25-28 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Affirmative Answer ◽  
Commutativity Theorem ◽  
Nilpotent Elements ◽  
Periodic Ring ◽  
Positive Integers

A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

2019 ◽  
Vol 18 (09) ◽  
pp. 1950167 ◽  
Author(s):  
M. Chacron ◽  
T.-K. Lee
Keyword(s):  
Finite Order ◽  
Division Ring ◽  
Affirmative Answer ◽  
Major Result ◽  
Division Rings ◽  
Open Questions ◽  
Engel Condition ◽  
Open Question ◽  
Positive Integers

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

Some new characterizations of periodic rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2050235 ◽  
Author(s):  
Jian Cui ◽  
Peter Danchev
Keyword(s):  
Periodic Ring ◽  
Periodic Rings ◽  
Positive Integers ◽  
Unital Rings ◽  
Regular Rings ◽  
Comprehensive Study

A ring [Formula: see text] is called periodic if, for every [Formula: see text] in [Formula: see text], there exist two distinct positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. The paper is devoted to a comprehensive study of the periodicity of arbitrary unital rings. Some new characterizations of periodic rings and their relationship with strongly [Formula: see text]-regular rings are provided as well as, furthermore, an application of the obtained main results to a ∗-version of a periodic ring is being considered. Our theorems somewhat considerably improved on classical results in this direction.

scholarly journals Generalized periodic rings

1996 ◽  
Vol 19 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Howard E. Bell ◽  
Adil Yaqub
Keyword(s):  
Opposite Parity ◽  
Periodic Ring ◽  
Periodic Rings ◽  
Positive Integers

LetRbe a ring, and letNandCdenote the set of nilpotents and the center ofR, respectively.Ris called generalized periodic if for everyx∈R\(N⋃C), there exist distinct positive integersm,nof opposite parity such thatxn−xm∈N⋂C. We prove that a generalized periodic ring always has the setNof nilpotents forming an ideal inR. We also consider some conditions which imply the commutativity of a generalized periodic ring.

scholarly journals A commutativity theorem for semi-primitive rings

1986 ◽  
Vol 34 (2) ◽  
pp. 293-295 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Commutativity Theorem ◽  
Positive Integers ◽  
Primitive Ring

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x, y ε R there exist positive integers m = m (x,y) and n = n (x,y) such that either [xm,(xy) n − (yx) n] = 0 or [xm,(xy) n + (yx) n] = 0. Then R is commutative.

scholarly journals A Generalization of Commutativity Theorem for Rings

2015 ◽  
Vol 61 (1) ◽  
pp. 97-100
Author(s):  
Junchao Wei ◽  
Zhiyong Fan
Keyword(s):  
Commutative Ring ◽  
Commutativity Theorem ◽  
Positive Integers

Abstract Let R be a ring with an identity and for each x ∈ SN(R) = {x ∈ R|x ∉ N(R)} and y ∈ R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring, which generalizes the known theorem belonging to Ligh and Richoux.

scholarly journals On Generalized Periodic-Like Rings

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Howard E. Bell ◽  
Adil Yaqub
Keyword(s):  
Jacobson Radical ◽  
Opposite Parity ◽  
Basic Properties ◽  
Nilpotent Elements ◽  
Positive Integers

LetRbe a ring with centerZ, Jacobson radicalJ, and setNof all nilpotent elements. CallRgeneralized periodic-like if for allx∈R∖(N∪J∪Z)there exist positive integersm,nof opposite parity for whichxm−xn∈N∩Z. We identify some basic properties of such rings and prove some results on commutativity.

scholarly journals A commutativity theorem for rings

1970 ◽  
Vol 2 (1) ◽  
pp. 95-99 ◽  
Author(s):  
D.L. Outcalt ◽  
Adil Yaqub
Keyword(s):  
Associative Ring ◽  
Commutativity Theorem ◽  
Nilpotent Elements

Let R be an associative ring with identity in which every element is either nilpotent or a unit. The following results are established. The set N of nilpotent elements in R is an ideal. If R/N is finite and if x ≡ y (mod N) implies x2 = y2 or both x and y commute with all elements of N, then R is commutative. Examples are given to show that R need not be commutative if “X2 = y2” is replaced by “xk = yK” for any integer k > 2. The case N = (0) yields Wedderburn's Theorem.

scholarly journals $1/k$-Eulerian Polynomials and $k$-Inversion Sequences

10.37236/8466 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Ting-Wei Chao ◽  
Jun Ma ◽  
Shi-Mei Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Affirmative Answer ◽  
Eulerian Polynomials ◽  
Bijective Proof ◽  
Positive Integers

Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-inversion sequence of length $n$ is a sequence ${\bf e} = (e_1, e_2, \ldots, e_n)$ of nonnegative integers such that $0 \leq e_i < s_i$ for $1\leq i\leq n$. When $s_i=(i-1)k+1$ for any $i\geq 1$, we call the ${\bf s}$-inversion sequences the $k$-inversion sequences. In this paper, we provide a bijective proof that the ascent number over $k$-inversion sequences of length $n$ is equidistributed with a weighted variant of the ascent number of permutations of order $n$, which leads to an affirmative answer of a question of Savage (2016). A key ingredient of the proof is a bijection between $k$-inversion sequences of length $n$ and $2\times n$ arrays with particular restrictions. Moreover, we present a bijective proof of the fact that the ascent plateau number over $k$-Stirling permutations of order $n$ is equidistributed with the ascent number over $k$-inversion sequences of length $n$.

scholarly journals A commutativity theorem for rings

1977 ◽  
Vol 16 (1) ◽  
pp. 75-77 ◽  
Author(s):  
Steve Ligh ◽  
Anthony Richoux
Keyword(s):  
Commutative Ring ◽  
Commutativity Theorem ◽  
Positive Integers

Let R be a ring with an identity and for each x, y in R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring.

scholarly journals Commutativity theorems for rings with constraints on commutators

1991 ◽  
Vol 14 (4) ◽  
pp. 683-688
Author(s):  
Hamza A. S. Abujabal
Keyword(s):  
Polynomial Identity ◽  
Nilpotent Elements ◽  
Zero Divisors ◽  
Commutativity Theorems ◽  
Associative Rings ◽  
Positive Integers

In this paper, we generalize some well-known commutativity theorems for associative rings as follows: Letn>1,m,s, andtbe fixed non-negative integers such thats≠m−1, ort≠n−1, and letRbe a ring with unity1satisfying the polynomial identityys[xn,y]=[x,ym]xtfor ally∈R. Suppose that (i)RhasQ(n)(that isn[x,y]=0implies[x,y]=0); (ii) the set of all nilpotent elements ofRis central fort>0, and (iii) the set of all zero-divisors ofRis also central fort>0. ThenRis commutative. IfQ(n)is replaced by “mandnare relatively prime positive integers,” thenRis commutative if extra constraint is given. Other related commutativity results are also obtained.

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