Let A be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A. In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals ...that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.
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.
We define an R-linear map φ from A to an A-bimodule M is said to be Jordan homo-derivation if φ(x2) = φ(x)x + xφ(x) + φ(x)2 for each x ∈ A. In this article, we proved that every Jordan ...homo-derivation to be homo-derivation on triangular algebras.
The present paper aims to study the connection between commutativity of rings and the behaviour of their generalized derivations. More precisely, we investigate the commutative prime rings
R
, which ...admit generalized derivations
Ψ
,
Φ
, and
Θ
satisfying specific differential identities on a certain subset of
R
.
This paper's primary goal is to look at a quotient ring $\mathscr{A}/\mathscr{T}$ structure, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. More ...precisely, we examine the differential identities in a semi-prime ideal of an arbitrary ring involving $\mathscr{T}$-commuting generalized derivation. Furthermore, examples are given to prove that the restrictions imposed on the hypothesisof the various theorems were not superfluous.
This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples ...to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems.
Let ℧ be a prime ring of char(℧) ≠2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity ...of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results.
This study aims to investigate the commutativity of a prime ring R with a non-zero ideal I and a homo-derivation ů that satisfies certain algebraic identities. We also provided some examples of why ...our results hypothesis is essential.
The purpose of this paper is to study the commutativity of a prime ring
with
anti-automorphism ψ and generalized derivation
satisfying certain algebraic
identities. Furthermore, we describe the ...structure of any additive map which is ψ-centralizing on
a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that
various conditions imposed on the hypotheses of our results are essential.
