Jordan Derivations Research Articles

Jordan derivations on Hilbert C*-modules

Assume that M is a Hilbert C ∗ -module over a C ∗ -algebra A and δ : M → M is a map. It is shown that if δ is A -linear, then the three statements are equivalent: (1) δ is a Jordan derivation; (2) δ is a derivation; (3) δ ∈ L ( M ) and δ ∗ = − δ . Furthermore, we give a decomposition of M and discuss some properties about this decomposition. Based on these, it is shown that, under the assumption that A is commutative, δ is a C -linear Jordan derivation if and only if δ is a derivation. Our results generalize some known related results.

CERTAIN TYPES OF DERIVATIONS IN RINGS: A SURVEY

In this overview article, we provide a historical account on derivations, Jordan derivations, (α, β)-derivations, left derivations, pre-derivations, homoderivations, nilpotent derivations, and other variants, drawing from the contributions of multiple researchers. Additionally, we delve into recent findings and suggest potential avenues for future investigation in this area. Furthermore, we offer pertinent examples to illustrate that the assumptions underlying various results are indeed necessary and not redundant.

Generalized Projective product of semi-rings

The concept of Differential algebra has been played an influential role in various directions of abstract algebra. This notation has been considered before fifty years ago with semi-ring and several types of rings. Furthermore, the commutativity of a given ring or semi-ring has been also investigated under the notation of derivation and many studies have been published in this way. Based on the concept of derivation, its generalization has been introduced by Bresar [9] and a lot of works are presented about this generalization in various types of rings and semi-rings. Motivated by these works in the literature, we introduced the concept of generalized projective product of n-number of semi-rings. In addition, we studied the convers of this case which is saying that if the generalized projective product of n-number of semi-rings existed, then there exist a corresponding derivations, generalized derivations, semi-derivations, generalized semi-derivations, Jordan derivations, generalized Jordan derivations, Jordan semi-derivations and Generalized Jordan semi-derivations on the semi-rings Ri respectively for any i=1,2,..,k.

Identities involving b-generalized skew derivations in prime rings acting as Jordan derivation

Let R symbolize a prime ring with a characteristic other than 2, and let ρ ( η 1 , … , η n ) represent a non-central multilinear polynomial over the extended centroid C of R . Assuming the presence of three b-generalized skew derivations on R , namely D 1 , D 2 , and D 3 that satisfy the equation D 1 ( ρ ) ρ + ρ D 2 ( ρ ) = D 3 ( ρ 2 ) for all ρ = ρ ( η 1 , … , η n ) , where η 1 , … , η n are elements of R . In this research paper, our goal is to elucidate all potential configurations of D 1 , D 2 , and D 3 .

Commuting Jordan Derivations on Triangular Rings Are Zero

Commuting Jordan Derivations on Triangular Rings Are Zero

Jordan derivations on triangular algebras and trivial extensions

In this paper, we provide a necessary and sufficient condition for a class of trivial extensions so that each Jordan derivation on such algebras can be written as the sum of a derivation and an antiderivation. Among other things, we obtain some of the known results concerning the Jordan derivations on trivial extensions with fewer assumptions. Also, as a generalization of a well-known result on triangular matrix algebras, we show that every Jordan derivation from arbitrary triangular algebra into its certain bimodule is the sum of a derivation and an antiderivation. We also apply our results for upper triangular matrix algebra [Formula: see text], matrix algebra [Formula: see text] and the nest algebra [Formula: see text] to describe the Jordan derivations into their bimodules.

Lie triple derivation and Lie bi-derivation on quaternion rings

In this study, we prove the existence of the central Lie bi-derivation for the ring with identity on the quaternion ring. We also describe the triple Lie derivation using the Jordan derivation on the aforementioned ring. An example is provided to demonstrate that our result is theoretically viable.

On Lie Derivations, Generalized Lie Derivations and Lie Centralizers of Octonion Algebras

Let L be a unital ring with characteristic different from 2 and O ( L ) be an algebra of Octonion over L . In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on O ( L ) . The matrix representation of Lie derivation of O ( L ) and its decomposition in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. The result about the decomposition of Lie centralizer of O in terms of Lie centralizer and Jordan centralizer of L is given. Moreover, the matrix representation of generalized Lie derivation (also known as D -Lie derivation) of O ( L ) is computed.

Functional identities on Banach algebras and matrix rings

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ ( A ) + δ ( A ) A − 1 + A τ ( A − 1 ) + τ ( A − 1 ) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ ( A ) = D ( A ) + A δ ( I ) and τ ( A ) = D ( A ) − δ ( I ) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ ( A ) B + A τ ( B ) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M . Similar conclusions are obtained for additive mappings from a matrix ring M n ( D ) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 .

On the Characterization of Skew Jordan Derivations in $$*$$-Rings

On the Characterization of Skew Jordan Derivations in $$*$$-Rings

On the Investigation of Derivation Pairs in Relation to Semi-Rings

The purpose of this article is to give some applications of the variations subordination concept on subclasses. The key objectives of the preceding paper are to investigate the dependency principle and to attempt to add a further group over polyvalent works via an extra operator related to derivative goods with a greater order. Several investigations were carried out to look into variations algebra musical notation by using algebraic frameworks like a ring or semi-ring for investigating some of its characteristics. The findings from the analysis of were additionally extended in this article through including the notation of derivation pair and studying some of its features. Furthermore, a couple of instances are provided to demonstrate that any derivation combination is a Jordan derivation combine but not vice versa.

On the derivations of the quadratic Jordan product in classical finite dimensional Cartan factors

Let R be a unital *-ring which contains a complex unit and in which the unit can be halved. Conmutativity of R is not assumed. Let Mn and Sn be the Jordan *-triples of the square n×n matrices (respectively, n×n symmetric matrices) (n≥2) with entries in R. The additive groups Der(Mn) and Der(Sn) of the quadratic Jordan derivations of Mn (respectively, Sn) are described in terms of Der(R), the group of quadratic Jordan derivations of R. Particular attention is paid to the case R=C.

Nonlinear bi-skew Jordan derivations in prime ∗-rings

Let [Formula: see text] be a unital prime ∗-ring containing a nontrivial symmetric idempotent and let [Formula: see text] be the maximal symmetric ring of quotients of [Formula: see text]. Using the technique of Peirce decomposition and the theory of functional identities, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if [Formula: see text] is an additive ∗-derivation unless [Formula: see text] and [Formula: see text]. As an application, we shall characterize such maps in different operator algebras.

Derivations characterized by monomials x2n in prime rings

Let [Formula: see text] be a prime ring of [Formula: see text] or [Formula: see text] and let [Formula: see text] be an additive map such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the maximal symmetric ring of quotients of [Formula: see text]. It is shown that there exist a derivation [Formula: see text] and an additive map [Formula: see text] with [Formula: see text] for all [Formula: see text], such that [Formula: see text]. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces.

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.

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 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.

A note on nonlinear mixed Jordan triple derivation on *-algebras

ABSTRACT Let be a unital *-algebra containing a non-trivial projection. In this paper, it is shown that a map Γ: such that for all . Then Γ is an additive *-derivation.

Nonlinear Jordan derivations of finitary incidence algebras

Let P be a partially ordered set, a 2-torsion free commutative ring with unity and the finitary incidence algebra of P over . In this paper, we give an explicit description for the structure of nonlinear Jordan derivations of . We show that if P has no trivial component, then every nonlinear Jordan derivation of is proper, and can be presented as a sum of a generalized inner derivation, a transitive induced derivation and an additive induced derivation. If P has a trivial connected component, we prove that every nonlinear Jordan derivation of is proper if and only if every nonlinear Jordan derivation of R is also proper.

JORDAN RIGHT DERIVATIONS ON SEMIPRIME Γ-RING

In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then . In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then .