On Jordan Ideals and Jordan Derivations of Prime Rings

Abstract

Abstract. Let R be a 2-torsion free prime ring, and let J be a nonzero Jordan ideal and a subring of R. In the present paper it is shown that if d is an additive mapping of IntroductionThroughout this paper R will denote an associative ring with center Z(R). Recall that R is prime if aRb -(0) implies that a = 0 or b = 0. As usual [x, y] and x o y will denote the commutator xy -yx and anticommutator xy + yx, respectively. A ring R is said to be 2-torsion free, if whenever 2x = 0, with x G R, then x = 0. An additive subgroup J of R is said to be a Jordan ideal of R if u o r € J, for all u G J, r € R. An additive mapping d : R -• R is called a derivation if d(xy) = d(x)y + xd(y), holds for all pairs x, y G R. An additive mapping d : R -> R is called a Jordan derivation if d(x 2 ) = d(x)x + xd(x)holds for all x G R. Obviously, every derivation on a ring R is a Jordan derivation. The converse is, in general, not true. A well known result due to Herstein [6] shows that every Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result is presented in [4]. This result was generalized by many authors (cf. [5]). In the present paper, our objective is to generalize this result for derivations defined on a subset of a prime rings. Preliminary resultsTo facilitate our discussion, we introduce abbreviation ¿(x, y) = d(xy) -d(x)y -xd(y). We shall make use of commutator identities:

Generalized (σ,τ) higher derivations in prime rings

“…A well-known result due to Herstein (2002) states that every Jordan derivation on a prime ring of characteristic different from two is a derivation. This result was further generalized by many authors in various directions (see Ashraf et al 2001; Bres̆ar and Vukman 1991where further references can be found). Motivated by these results Ferrero and Haetinger (2002) generalized Herstein’s theorem for higher derivations and proved that every Jordan higher derivation on a prime ring of characteristic different from two is a higher derivation.…”

“…We’ll proceed by induction on n . For n = 1, every generalized Jordan ( σ , τ )-higher derivation reduces to generalized Jordan ( σ , τ )-derivation and hence using Lemma 2.2( iii ) of (Ashraf et al 2001), we have …”

On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

“…A famous result due to Bre²ar [5,Theorem 4.3], asserts that a Jordan triple derivation on a 2-torsion free semiprime ring is a derivation. Following the same line, a number of results have been obtained by several authors (see [2], [3], [4], [9], [18], [19], [22], [23]), where further references can be found.…”

Jordan \alpha-centralizers in rings and some applications

“…Following B. Hvala [ [9], page 1447], an additive mapping F : R → R is called a generalized derivation if there exists a derivation d: R → R such that F (xy) = F (x)y + xd(y) holds for all x, y ∈ R. We call an additive mapping F : R → R a generalized Jordan derivation if there exists a derivation d: R → R such that F (x 2 ) = F (x)x + xd(x) holds for all x ∈ R [ [5], page 7]. In [ [5], Theorem], M. Ashraf and N. Rehman showed that in a 2-torsion-free ring R which has a commutator nonzero divisor, every generalized Jordan derivation on R is a generalized derivation.…”