On n-generalized commutators and Lie ideals of rings

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.

On the subring generated by commutators

Let [Formula: see text] be a ring. By the notation [Formula: see text] we denote the additive subgroup of [Formula: see text] generated by all [Formula: see text] in [Formula: see text]. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if [Formula: see text], then the subring generated by [Formula: see text] is equal to [Formula: see text]. This result implies that [Formula: see text] cannot be a proper subring of [Formula: see text].

(σ,τ)-*-Jordan ideals in *-prime rings

Let R be a prime ring with involution {\star} , and let σ, τ be endomorphisms on R. For any {x,y\in R} , let {(x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x} and {C_{\sigma,\tau}(R)=\{x\in R\mid x\sigma(y)=\tau(y)x\}} . An additive subgroup U of R is said to be a {(\sigma,\tau)} -right Jordan ideal (resp. {(\sigma,\tau)} -left Jordan ideal) of R if {(U,R)_{\sigma,\tau}\subseteq U} (resp. {(R,U)_{\sigma,\tau}\subseteq U} ), and U is called a {(\sigma,\tau)} -Jordan ideal if U is both a {(\sigma,\tau)} -right Jordan ideal and a {(\sigma,\tau)} -left Jordan ideal of R. A {(\sigma,\tau)} -Jordan ideal U of R is said to be a {(\sigma,\tau)} - {\star} -Jordan ideal if {U^{\star}=U} . In the present paper, it is shown that if U is commutative, then R is commutative. The commutativity of R is also obtained if {(U,U)_{\sigma,\tau}\subseteq C_{\sigma,\tau}(R)} . Some more results are obtained on the {\star} -prime ring with a characteristic different from 2.

On Asano’s theorem

Let [Formula: see text] be a division algebra over an infinite field [Formula: see text] such that every element of [Formula: see text] is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of [Formula: see text] invariant under all inner automorphisms induced by algebraic elements contains [Formula: see text], the additive subgroup of [Formula: see text] generated by all additive commutators of [Formula: see text]. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that [Formula: see text] is generated by multiplicative commutators as a vector space over the center of [Formula: see text].

Rings which are sums of PI subrings

We study rings [Formula: see text] which are sums of a subring [Formula: see text] and an additive subgroup [Formula: see text]. We prove that if [Formula: see text] is a prime radical ring and [Formula: see text] satisfies a polynomial identity, then [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. Additionally, we show that if [Formula: see text] is a [Formula: see text] ring, then the prime radical of [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. We also obtain a new condition equivalent to Koethe’s conjecture.

Some notes on Lie ideals in division rings

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

Central Extensions and Derivations of Generalized Schrödinger-Virasoro Algebras

Let 𝔽 be a field of characteristic 0, G an additive subgroup of 𝔽, s ∈ 𝔽 such that s ∉ G and 2s ∈ G. A class of infinite-dimensional Lie algebras [Formula: see text] called generalized Schrödinger-Virasoro algebras was defined by Tan and Zhang, which is a natural generalization of Schrödinger-Virasoro algebras. In this paper, central extensions and derivations of [Formula: see text] are determined.

Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.