scholarly journals On commutativity in certain rings

1970 ◽  
Vol 2 (1) ◽  
pp. 107-115 ◽  
Author(s):  
H.G. Moore
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Associative Ring ◽  
Arbitrary Element ◽  
Alternative Ring ◽  
Commutator Ideal ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Fixed Positive Integer ◽  
Associative Rings

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.

scholarly journals Rings whose additive endomorphisms are N-multiplicative

1989 ◽  
Vol 39 (1) ◽  
pp. 11-14 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Homogeneous Polynomial ◽  
Integer Coefficients ◽  
Additive Endomorphism ◽  
Fixed Positive Integer

Sullivan's problem of describing rings, all of whose additive endomorphisms are multiplicative, is generalised to the study of rings R satisfying ϕ(a1 … an) = ϕ(a1)…ϕ(an) for every additive endomorphism ϕ of R, and all a1,…,an ∈ R, with n > 1 a fixed positive integer. It is shown that such rings possess a bounded (finite) ideal A such that [R/A]n = 0 ([R/A]2n−1 = 0). More generally, if f(X1, …, Xt) is a homogeneous polynomial with integer coefficients, of degree > 1, and if a ring R satisfies ϕ[f(a1, …, at)] = f[ϕ(a1), …, ϕ(at)] for all additive endomorphisms ϕ, and all a1, …, at ∈ R, then R possesses a bounded ideal A such that R/A satisfies the polynomial identity f.

scholarly journals Commutativity results for rings

1988 ◽  
Vol 38 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Hazar Abu-Khuzam
Keyword(s):  
Positive Integer ◽  
Finite Subset ◽  
Associative Ring ◽  
Free Ring ◽  
Commutator Ideal ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

scholarly journals Ideals in simple rings

1978 ◽  
Vol 25 (3) ◽  
pp. 322-327
Author(s):  
W. Harold Davenport
Keyword(s):  
Lie Algebra ◽  
Associative Ring ◽  
Alternative Ring ◽  
Cayley Algebra ◽  
Characteristic 3 ◽  
Associative Rings ◽  
Algebra Over A Field

AbstractIn this article, we define the concept of a Malcev ideal in an alternative ring in a manner analogous to Lie ideals in associative rings. By using a result of Kleinfield's we show that a nonassociative alternative ring of characteristic not 2 or 3 is a ring sum of Malcev ideals Z and [R, R] where Z is the center of R and [R, R] is a simple non-Lie Malcev ideal of R. If R is a Cayley algebra over a field F of characteristic 3 then [R, R] is a simple 7 dimensional Lie algebra. A similar result is obtained if R is a simple associative ring.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

scholarly journals On transposes of nilpotent matrices over arbitrary rings

1977 ◽  
Vol 23 (3) ◽  
pp. 366-370 ◽  
Author(s):  
Thomas P. Kezlan
Keyword(s):  
Division Ring ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Commutator Ideal ◽  
Nilpotent Elements ◽  
Nilpotent Matrices

AbstractIt is shown that if every nilpotent 2 × 2 matrix over a ring has nilpotent transpose, then the commutator ideal must be contained in the Jacobson radical, thus generalizing a result of R. S. Gupta, who considered the division ring case. Moreover, if the nilpotent elements form an ideal or if the ring satisfies a polynomial identity, then the above property of the transpose implies that in fact the commutator ideal must be nil.

scholarly journals On the Power Map and Ring Commutativity

1978 ◽  
Vol 21 (4) ◽  
pp. 399-404 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Commutator Ideal ◽  
Image Position

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identitiesfor all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.

scholarly journals Leavitt path algebras with bounded index of nilpotence

2019 ◽  
Vol 18 (10) ◽  
pp. 1950185 ◽  
Author(s):  
Kulumani M. Rangaswamy ◽  
Ashish K. Srivastava
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Oxford University ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Fixed Positive Integer ◽  
Index Of Nilpotence ◽  
Open Question

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra [Formula: see text] has index of nilpotence at most [Formula: see text] if and only if no cycle in the graph [Formula: see text] has an exit and there is a fixed positive integer [Formula: see text] such that the number of distinct paths that end at any given vertex [Formula: see text] (including [Formula: see text], but not including the entire cycle [Formula: see text] in case [Formula: see text] lies on [Formula: see text]) is less than or equal to [Formula: see text]. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial identity. Furthermore, Leavitt path algebras with bounded index of nilpotence are shown to be directly-finite and to be [Formula: see text]-graded [Formula: see text]–[Formula: see text] rings. As an application of our results, we answer an open question raised in [S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs (Oxford University Press, 2012)] whether an exchange [Formula: see text]–[Formula: see text] ring has bounded index of nilpotence.

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.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

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