Differential identities in prime rings

Let R be a prime ring with center Z(R) and alpha,beta be automorphisms of R. This paper is divided into two parts. The first tackles the notions of (generalized) skew derivations on R, as the subject of the present study, several characterization theorems concerning commutativity of prime rings are obtained and an example proving the necessity of the primeness hypothesis of R is given. The second part of the paper tackles the notions of symmetric Jordan bi (alpha,beta)-derivations. In addition, the researchers illustrated that for a prime ring with char(R) different from 2, every symmetric Jordan bi (alpha,alpha)-derivation D of R is a symmetric bi (alpha,alpha)-derivation.

Some results on ∗-differential identities in prime rings

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

On Jordan ideals and Generalized (α, 1)- Reverse derivations in ∗-prime rings

Let R will be a 2- torsion free ∗-prime ring and α be an automorphisum of R. F be a nonzero generalized (α, 1)- reverse derivation of R with associated nonzero (α, 1)- reverse derivation d which commutes with ∗ and J be a nonzero ∗-Jordan ideal and a subring of R. In the present paper, we shall prove that R is commutative if any one of the following holds: (i) [F(u), u]α,1 = 0, (ii) F(u) α(u) = ud(u), (iii) F(u2) = ± α(u2), (iv) F(u2) = 2d(u) α(u), (v) d(u2) = 2F(u) α(u), for all u ∈ U.

On Commutativity of Prime Rings with Symmetric Left θ-3- Centralizers

Let R be an associative ring with center Z(R) , I be a nonzero ideal of R and be an automorphism of R . An 3-additive mapping M:RxRxR R is called a symmetric left -3-centralizer if M(u1y,u2 ,u3)=M(u1,u2,u3)(y) holds for all y, u1, u2, u3 R . In this paper , we shall investigate the commutativity of prime rings admitting symmetric left -3-centralizer satisfying any one of the following conditions : (i)M([u ,y], u2, u3) [(u), (y)] = 0 (ii)M((u ∘ y), u2, u3) ((u) ∘ (y)) = 0 (iii)M(u2, u2, u3) (u2) = 0 (iv) M(uy, u2, u3) (uy) = 0 (v) M(uy, u2, u3) (uy) For all u2,u3 R and u ,y I

Generalized g-derivations on prime rings

We introduce the definitions of [Formula: see text]-derivations and generalized [Formula: see text]-derivations on a ring [Formula: see text]. The main objective of the paper is to describe the structure of a prime ring [Formula: see text] in which [Formula: see text]-derivations and generalized [Formula: see text]-derivations satisfy certain algebraic identities with involution ⋆, anti-automorphism and automorphism. Some well-known results concerning derivations, generalized derivations, skew derivations and generalized skew derivations in prime rings, have been generalized to the case of [Formula: see text]-derivations and generalized [Formula: see text]-derivations.