A Note on Multiplicative (Generalized) (α, β)-Derivations in Prime Rings

Abstract

Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

“…Taking H by H − 1 and G by G + 1 in (3) and then using Teorem 6 (2), we get (3). Taking H by H − 1 in (4) and then using (3), we get (4). Taking G by G − 1 in (5) and then using (3), we get (5).…”

“…Te ideas of multiplicative centralizers are covered by a multiplicative (generalized)-derivation associated with mapping ψ � 0 (not necessarily additive). However, there are few articles on this subject (see [3][4][5] for a partial bibliography).…”

Remarks on the notion of homo-derivations

“…In [4], El Sofy introduced the notion of homo-derivations. After that several results appeared where the authors proved commutativity results for the domain of these mappings, see e. g. [1,2,8,11]. It is objectionable however that there have not been made attempt to characterize or to compare these notions.…”

On Multiplicative (Generalized)-Derivations and Central Valued Conditions in Prime Rings

“…During the last six decades there have been many results showing that the global structure of a ring is often tightly connected with the behaviour of additive and multiplicative mappings defined on it (see [2], [9], [10], [11], [12], [13]). In 1957, Posner [11] initiated the study of identities involving derivations that ensure commutativity.…”