scholarly journals A note on rings with certain variables identities

1989 ◽  
Vol 12 (3) ◽  
pp. 463-466
Author(s):  
Hazar Abu-Khuzam
Keyword(s):  
Commutator Ideal ◽  
Generalized Commutator

It is proved that certain rings satisfying generalized-commutator constraints of the form[xm,yn,yn,...,yn]=0with m and n depending on x and y, must have nil commutator ideal.

scholarly journals Some commutativity results for rings

1980 ◽  
Vol 22 (2) ◽  
pp. 285-289 ◽  
Author(s):  
Abraham A. Klein ◽  
Itzhak Nada ◽  
Howard E. Bell
Keyword(s):  
Commutator Ideal ◽  
Generalized Commutator

It is proved that certain rings satisfying generalized-commutator constraints of the form [xm, yn, yn, …, yn] = 0 must have nil commutator ideal.

scholarly journals The Nowicki conjecture for free metabelian Lie algebras

2019 ◽  
Vol 19 (05) ◽  
pp. 2050095
Author(s):  
Vesselin Drensky ◽  
Şehmus Fındık
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Algebra ◽  
Vector Subspace ◽  
Classical Theorem ◽  
Finitely Generated ◽  
Commutator Ideal ◽  
Ideal Formula ◽  
Locally Nilpotent ◽  
Derivation Formula

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

scholarly journals A note on subdirectly irreducible rings

1971 ◽  
Vol 4 (1) ◽  
pp. 31-34
Author(s):  
Madhukar G. Deshpande
Keyword(s):  
Homomorphic Image ◽  
Integral Domain ◽  
Subdirectly Irreducible ◽  
Commutator Ideal

It is proved that every subdirectly irreducible ring can be obtained as a proper homomorphic image of another subdirectly irreducible ring. An example of a subdirectly irreducible ring R with heart H is given for which i) R has a non-subdirectly irreducible homomorphic image, ii) R/H is an integral domain, and iii) the commutator ideal C(R) of R coincides with R.

On Rings With Engel Cycles

1991 ◽  
Vol 34 (3) ◽  
pp. 295-300 ◽  
Author(s):  
H. E. Bell ◽  
A. A. Klein
Keyword(s):  
Commutator Ideal ◽  
Positive Integers

AbstractA ring R is called an EC-ring if for each x, y ∊ R, there exist distinct positive integers m, n such that the extended commutators [x, y]m and [x, y]n are equal. We show that in certain EC-rings, the commutator ideal is periodic; we establish commutativity of arbitrary EC-domains; we prove that a ring R is commutative if for each x, y ∊ R, there exists n > 1 for which [x, y] = [x, y]n.

On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

2010 ◽  
Vol 17 (04) ◽  
pp. 685-698 ◽  
Author(s):  
Shuan-hong Wang ◽  
Hai-xing Zhu
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Weak Hopf Algebra ◽  
Commutator Ideal ◽  
Commutative Subalgebras

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Generalized Commutator Formulas

2011 ◽  
Vol 39 (4) ◽  
pp. 1441-1454 ◽  
Author(s):  
R. Hazrat ◽  
Z. Zhang
Keyword(s):  
Generalized Commutator
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