Prime Rings Research Articles

On Lie ideals and multiplicative generalized (α, β)-reverse derivations of *-prime rings

In this paper, the notion of multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations of *-prime rings is characterized. The action of these derivations on *-Lie ideals of *-prime rings is also investigated. Moreover, the commutativity of *-prime rings admitting multiplicative generalized (α, β)-reverse derivations associated with (α, β)-reverse derivations satisfying certain algebraic identities on *-Lie ideals is explored.

On nilpotent homoderivations in prime rings

Let R be an associative ring and let be a fixed integer. An additive map h on R is called a homoderivation if holds for all In 1978, Herstein proved that a prime ring R of is commutative if there is a nonzero derivation d of R such that for all . The main objective of this paper is to prove the above mentioned result for homoderivations with nilpotency ‘s’ in prime rings. Moreover, we prove that if a prime ring admits homoderivations h1 and h2 such that and for positive integers s1 and s2, then at least one of the homoderivations must be nilpotent.

A Characterization of Semiprime Rings with Homoderivations

This paper is focused on the commutativity of the laws of semiprime rings, which satisfy some algebraic identities involving homoderivations on ideals. It provides new and notable results that will interest researchers in this field, such as “R contains a nonzero central ideal if R admits a nonzero homoderivation δ on I such that δ(I)⊆Z where R is a semiprime ring with center Z and I a nonzero ideal of R”. Moreover, the research also generalizes some results previously published in the literature, including derivation on prime rings using homoderivation semiprime rings. It also demonstrates the necessity of hypotheses operationalized in theorems by an example. Finally, the paper discusses how the results herein can be further developed in future research.

A note on certain additive maps in prime rings with involution

A note on certain additive maps in prime rings with involution

Strong 3-skew commutativity preserving maps on prime rings with involution

ABSTRACT Let be a unital prime -ring containing a nontrivial symmetric idempotent. For , the 3-skew commutator is defined by . Let be a surjective map. It is shown that satisfies for all if and only if there exists with such that for all . Where I is the unit of and is the symmetric center of . This result then is applied to some operator algebras. Communicated by Igor Klep

Automorphisms with Lie İdeals and (η,ξ)-derivations

Let S be a prime ring with charS≠2 and η, ξ, κ, ε, λ and μ will be automorphisms of S. The purpose of our paper is 6 to study (η,ξ)-derivations and automorphisms satisfying certain identities on Lie ideals of prime rings.

On generalized derivations and commutativity of prime rings with involution

In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on commutativity of prime rings have been obtained. Also, we provide an example to show that the assumed restriction imposed on the involution is not superfluous.

On centrally extended Jordan derivations and related maps in rings

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $\delta:R\to R$ such that\[[\delta(x),x]\in Z(R)~~(resp.~~[\delta(x),x^{\ast}]\in Z(R))\text{ for all }x\in R,\] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.

On noncommutative bounded factorization domains and prime rings

A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is a bound λ(a) on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

B′-Generalized skew derivations acting as a Jordan derivation on multilinear polynomials in prime rings

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, C the extended centroid of R and a noncentral multilinear polynomial of R. Let F, H be -generalized skew derivations on R, with associated terms and respectively, and α commutes with d and δ. In this article we study the complete structure of F and H in the case , for all .

Centrally Extended Jordan (∗)-Derivations Centralizing Symmetric or Skew Elements

Let A be a non-commutative prime ring with involution ∗, of characteristic ≠2(and3), with Z as the center of A and Π a mapping Π:A→A such that [Π(x),x]∈Z for all (skew) symmetric elements x∈A. If Π is a non-zero CE-Jordan derivation of A, then A satisfies s4, the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A, then A satisfies s4 or Π(y)=λ(y−y*) for all y∈A, and some λ∈C, the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results.

On two commutativity criteria for δ-prime rings

The note concerns the commutativity of associative rings (possibly nonunital) endowed with a derivation. Our focus is on ?-prime rings. We give a new proof of Hirano and Tominaga?s result that a ?-prime ring is commutative whenever the derivation ? is nonzero and commuting on a nonzero two-sided ?-ideal. We also provide some further generalizations of Herstein?s classical theorem on a prime ring admitting a nonzero derivation with commutative range.

On Prime Ring with Reverse (𝛂, 𝛃)-Derivations

Throughout this research paper,R represent a prime ring. An additive mapping d∶ R→ R is said to be a derivation if it satisfies d(xy) = d(x)y + xd(y) for all x,y ∈R, alsod∶ R→ R is called a reverse derivation if it satisfies d(xy) = d(y)x + yd(x) for all x,y ∈R. A mapping F:R→R is said to be generalized derivation associated with derivation d∶ R→ R if F(xy)=F(x)y+xd(y) for all x,y ∈R and F:R→R is called a generalized reverse derivation associated with reverse derivation d:R→R if for all x,y∈R, thenF(xy)=F(y)x+yd(x).An additive mapping d: R →R associated with an automorphism α,β is called a reverse (α,β)-derivation on R if satisfying (xy)=d(y)α(x)+ β(y)d(x) , for all x,y∈R.The aim of this research paper is to establish the commutativity results of prime ring R by reverse (α,β)-derivation which satisfying certain constraints.

Centrally-extended generalized Jordan derivations in rings

Abstract In this article, we introduce the notion of centrally-extended generalized Jordan derivations and characterize the structure of a prime ring (resp. *-prime ring) R that admits a centrally-extended generalized Jordan derivation F satisfying [F(x), x] ∈ Z(R) (resp. [F(x), x*] ∈ Z(R)) for all x ∈ R.

An Engel condition with b-generalized derivations in prime rings

An Engel condition with b-generalized derivations in prime rings

Generalized (σ,τ)-reverse derivations in prime rings

Generalized (σ,τ)-reverse derivations in prime rings

Herstein's theorem for pair of generalized derivations on prime rings with involution

Herstein's theorem for pair of generalized derivations on prime rings with involution

Action of $ n $-derivations and $ n $-multipliers on ideals of (semi)-prime rings

<abstract><p>The present paper aims to investigate the containment of nonzero central ideal in a ring $ \mathcal{R} $ when the trace of symmetric $ n $-derivations satisfies some differential identities. Lastly, we prove that in a prime ring $ \mathcal{R} $ of suitable torsion restriction, if $ \mathfrak{D}, \mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R} $ are two nonzero symmetric $ n $-derivations such that $ \mathcal{f}(\vartheta)\vartheta +\vartheta\mathcal{g}(\vartheta) = 0 $ holds $ \forall \; \vartheta \in \mathcal{W} $, a nonzero left ideal of $ \mathcal{R} $ where $ \mathcal{f} $ and $ \mathcal{g} $ are the traces of $ \mathfrak{D} $ and $ \mathcal{G} $, respectively, then either $ \mathcal{R} $ is commutative or $ \mathcal{G} $ acts as a left $ n $-multiplier. Finally, we characterize symmetric $ n $-derivations in terms of left $ n $-multipliers.</p></abstract>

Additive mappings satisfying certain algebraic equations in prime rings

Additive mappings satisfying certain algebraic equations in prime rings

Commutativity of prime rings with symmetric generalized biderivations

Commutativity of prime rings with symmetric generalized biderivations