RINGS SATISFYING GENERALIZED ENGEL CONDITIONS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250121 ◽  
Author(s):  
M. RAMEZAN-NASSAB ◽  
D. KIANI
Keyword(s):  
Soluble Group ◽  
Associative Ring ◽  
Commutator Ideal ◽  
Engel Group ◽  
Algebra Over A Field

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0) and [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)]r(x, y) = 0). Among other results, it is proved that every Artinian [Formula: see text]-ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a [Formula: see text]-ring with unity and k, n independent of y; (2) if R is a locally bounded [Formula: see text]-ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.

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.

Third-Engel 2-Groups are Soluble

1972 ◽  
Vol 15 (4) ◽  
pp. 523-524 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Soluble Group ◽  
Engel Group

AbstractIt is shown that a 3rd-Engel group is an extension of a soluble group by a group of exponent 5.

scholarly journals Right alternative algebras with commutators in a nucleus

1992 ◽  
Vol 46 (1) ◽  
pp. 81-90
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Linear Span ◽  
Alternative Algebra ◽  
Wedderburn Decomposition ◽  
Alternative Algebras ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
The Right ◽  
Nil Radical ◽  
Algebra Over A Field

Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

scholarly journals On Reversible Group Rings

2006 ◽  
Vol 74 (1) ◽  
pp. 139-142 ◽  
Author(s):  
Yuanlin Li ◽  
Howard E. Bell ◽  
Colin Phipps
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Group Algebra ◽  
Associative Ring ◽  
Group Rings ◽  
Finite Rings ◽  
Arbitrary Finite Group ◽  
Algebra Over A Field

Let G be an arbitrary finite group, R be a finite associative ring with identity and RG be the group ring. We show that ℤ2Q8 is the minimal reversible group ring which is not symmetric, and we also characterise the finite rings R for which RQ8 is reversible. The first result extends a result of Gutan and Kisielewicz which shows that ℤ2Q8 is the minimal reversible group algebra over a field which is not symmetric, and it answers a question raised by Marks for the group ring case.

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 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 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.

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

On 4-Engel Groups

2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Normal Closure ◽  
Engel Group

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group

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.

Sign in / Sign up

Export Citation Format

Share Document