On Generalized Jordan Triple -Higher Derivations in Prime Rings

Let be a ring and let be a Lie ideal of . Suppose that are endomorphisms of , and is the set of all nonnegative integers. A family of mappings is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of if there exists a -higher derivation of such that , the identity map on , , and (resp., hold for all and for every If the above conditions hold for all , then is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of into . In the present paper it is shown that if is a noncentral square closed Lie ideal of a prime ring of characteristic different from two, then every generalized Jordan triple -higher derivation of into is a generalized -higher derivation of into . 1. Introduction Let be an associative ring with center (may be without identity element). For any , will denote the usual Lie product and the element will be denoted by . An additive subgroup of is said to be a Lie ideal of if . A Lie ideal of is said to be square closed Lie ideal of if for all . An additive mapping is said to be a derivation on if for all . Also, an additive mapping is called a Jordan triple derivation if for all . Herstein [1, Lemma 3.5] showed that on a 2-torsion free ring every derivation is Jordan triple derivation, but the converse need not be true in general, whereas Bre？ar [2] proved that on a 2-torsion free semiprime ring every Jordan triple derivation is a derivation. The concept of derivations was further extended to -derivation. Let be the endomorphisms of . An additive mapping is said to be a -derivation of if holds for all . Moreover, an additive mapping is said to be a Jordan triple -derivation of if holds for all . Bre？ar [3] introduced the notion of generalized derivation as follows: an additive mapping is said to be a generalized derivation on if there exists a derivation on such that for all . An additive mapping is said to be a generalized -derivation on if there exists a -derivation on such that for all (for reference see [4]). Correspondingly, the concept of generalized Jordan triple derivations was defined by Wu and Lu [5]. An additive mapping is said to be a generalized Jordan triple derivation on if there exists a Jordan triple derivation on such that for all . They also showed that every generalized Jordan triple derivation on a 2-torsion free prime ring is a generalized derivation. Further, Liu and Shiue [6] extended this result for generalized Jordan triple -derivation. An additive mapping is said to be a generalized Jordan triple -derivation on if there exists