Lie Ideals Research Articles

Annihilators of power values of generalized skew derivations on Lie ideals in prime rings

Annihilators of power values of generalized skew derivations on Lie ideals in prime rings

Generalized derivations with powercentral values on Lie ideals and Banach Algebras

Generalized derivations with powercentral values on Lie ideals and Banach Algebras

Engel condition of composition of generalized derivations in prime rings

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]. Recently, [Dhara et al., Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48(1) (2020) 154–167] studied the case [Formula: see text] for all [Formula: see text], where [Formula: see text] is a derivation (which is independent of [Formula: see text]) and [Formula: see text] fixed positive integers and obtained the structure of [Formula: see text] and the ring [Formula: see text]. [Liu, An Engel condition with two generalized derivations on Lie ideals of prime rings, J. Algebra and Its Applications, 22(5) (2023) 2350115] studied a similar identity replacing derivation [Formula: see text] with another generalized derivation [Formula: see text]. Recently, [Pandey, Pair of generalized derivations on Lie ideals in prime rings, Asian-European J. Math. 16(1) (2022) 2350002] studied an identity considering composition of mapping, that is [Formula: see text] for all [Formula: see text]. In this paper, we consider the situation containing composition of mappings as well as Engel condition, that is, [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed integers and then obtain the nature of mappings [Formula: see text], [Formula: see text] and the ring [Formula: see text].

Notes on multiplicative generalized (σ, τ )-reverse derivations with Lie ideals of semiprime ∗-rings

This article is devoted to investigation into the notion of multiplicative generalized(σ,τ)-reverse derivations associated with(σ,τ)-reverse derivations of semiprime∗-rings is characterized. The action of these derivations on∗-Lie ideals of semiprime∗-rings is also consideration. Moreover, the commutativity of semiprime∗-rings admitting multiplicative generalized(σ,τ)-reverse derivations associated with(σ,τ)-reverse derivations satisfying certain algebraic identitieson∗-Lie ideals is explored.

Hypercommuting conditions of b-generalized skew derivations on Lie ideals in prime rings

Hypercommuting conditions of b-generalized skew derivations on Lie ideals in prime rings

Lie Ideals with Multiplicative Homomultipliers on Near-Rings

In this study, the new types of mappings on the rings and near-rings are defined, which are named multiplicative homoleft multipliers, multiplicative homoright multipliers and multiplicative homo multipliers. We also present examples that show the existence of such mappings. Moreover, we study how these mappings effected on the construction of near-rings.

Prime ideals in C*-algebras and applications to Lie theory

We show that every proper, dense ideal in a C ∗ C^{*} -algebra is contained in a prime ideal. It follows that a subset generates a C ∗ C^{*} -algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings to C ∗ C^{*} -algebras. For example, if a C ∗ C^{*} -algebra A A is generated by its commutator subspace [ A , A ] [A,A] as a ring, then [ [ A , A ] , [ A , A ] ] = [ A , A ] [[A,A],[A,A]] = [A,A] . Further, given Lie ideals K K and L L in A A , then [ K , L ] [K,L] generates A A as a not necessarily closed ideal if and only if [ K , K ] [K,K] and [ L , L ] [L,L] do, and moreover this implies that [ K , L ] = [ A , A ] [K,L]=[A,A] . We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a C ∗ C^{*} -algebra.

$b$-generalized skew derivations acting on Lie ideals in prime rings

$b$-generalized skew derivations acting on Lie ideals in prime rings

Commutators and generalized derivations acting on Lie ideals in prime rings

Commutators and generalized derivations acting on Lie ideals in prime rings

Lie Modules of Banach Space Nest Algebras

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A-module. We show that there exist a weakly closed A-bimodule K, a weakly closed subalgebra DK of A, and a largest weakly closed A-bimodule J contained in L,such that J⊆L⊆K+DK, with [K,A]⊆L. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Leavitt Path Algebras in Which Every Lie Ideal is an Ideal and Applications

Leavitt Path Algebras in Which Every Lie Ideal is an Ideal and Applications

Power central valued b-generalized skew derivations on Lie ideals

Power central valued b-generalized skew derivations on Lie ideals

Homoderivations and Their Impact on Lie Ideals in Prime Rings

Assume we have a prime ring denoted as $R$, with a characteristic distinct from two. The concept of a homoderivation refers to an additive map $Η$ of a ring $R$ that satisfies the property $Η(r_1 r_2 )=Η(r_1 ) r_2+r_1 Η(r_2 )+Η(r_1 )Η(r_2 )$, $\forall r_1,r_2 \in R$. This article aims to obtain results for prime rings, ideals, and Lie ideals by utilizing the concept of homoderivation in conjunction with the established theory of derivations.

Identities with generalized derivations on Lie ideals and Banach algebras

Abstract Let 𝑅 be a prime ring and 𝐿 a non-central Lie ideal of 𝑅. In this paper, we aim to classify the generalized derivations of 𝑅 satisfying some algebraic identities with power values on 𝐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.

Results on Lie ideals of prime rings with homoderivations

Let R be a prime ring of characteristic not 2 and U be a noncentral square closed Lie ideal of R. An additive mapping H on R is called a homoderivation if H(xy) = H(x)H(y)+H (x) y+xH(y) for all x, y ∈ R. In this paper we investigate homoderivations satisfying certain differential identities on square closed Lie ideals of prime rings.

Product of generalized derivations with Engel condition on Lie ideals in prime rings

This paper represents the structures of generalized derivations satisfying an identity with [Formula: see text]-Engel condition. To prove our result, we take [Formula: see text] as a prime ring with [Formula: see text], and [Formula: see text] as Utumi quotient ring of [Formula: see text]. The center of [Formula: see text] is [Formula: see text] which is called extended centroid of [Formula: see text]. Let [Formula: see text] be a non-central Lie ideal of [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are two nonzero generalized derivations of [Formula: see text] and [Formula: see text], any fixed integer. If [Formula: see text] for all [Formula: see text], then one of the below-mentioned conclusions holds: (1) [Formula: see text] and [Formula: see text] for all [Formula: see text] and for some [Formula: see text] with [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text].

Generalized Skew Derivations with Hypercommuting Conditions on Lie Ideals

Generalized Skew Derivations with Hypercommuting Conditions on Lie Ideals

Identities related to a pair of generalized skew derivations on Lie ideals

Identities related to a pair of generalized skew derivations on Lie ideals

Generalized derivations with nilpotent values on Lie ideals in semiprime rings

Generalized derivations with nilpotent values on Lie ideals in semiprime rings