Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals

Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings. 1. Introduction and Preliminaries Throughout this paper, will denote an associative ring with center . We will write, for all , and for the Lie product and Jordan product, respectively. is -torsion free, if whenever , with , then . is prime if implies that or . If admits an involution , then is -prime if yields or . Note that every prime ring having an involution is -prime, but the converse is in general not true. Indeed, if denotes the opposite ring of a prime ring , then equipped with the exchange involution , defined by , is -prime but not prime. This example shows that every prime ring can be injected in a -prime ring, and from this point of view -prime rings constitute a more general class of prime rings. An additive mapping is a derivation on if for all . Let be a fixed element. A map defined by , , is a derivation on , which is called inner derivation defined by . Many results in the literature indicate how the global structure of a ring is often tightly connected to the behaviour of additive mappings defined on . A well-known result of Posner [1] states that if is a derivation of the prime ring such that , for any , then either or is commutative. In [2], Lanski generalizes the result of Posner to a Lie ideal. More recently, several authors consider similar situation in the case that the derivation is replaced by a generalized derivation. More specifically, an additive map is said to be a generalized derivation if there exists a derivation of such that, for all , . Basic examples of generalized derivations are the usual derivations on and left -module mappings from into itself. An important example is a map of the form , for some ; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore, any investigation from the algebraic point of view might be interesting (see, e.g., [3, 4]). In [5] Bergen introduced the notion of a semiderivation of a ring as follows: an additive mapping of into itself is called a semiderivation if there exists a function such that for all in . In case is the identity map on , is a derivation. Moreover, if is an automorphism of , is called skew derivation (or -derivation). Basic examples of -derivations are