GENERALIZED JORDAN TRIPLE HIGHER DERIVATIONS ON SEMIPRIME RINGS

Abstract

In this paper we prove that every generalized Jordan triple higher derivation on a 2-torsion free semiprime ring is a generalized higher derivation. This extend the main result of (9) to the case of a semiprime ring. 1. Introduction and preliminaries Let R be an associative ring not necessarily with identity element. A deriva- tion (resp. Jordan derivation) of R is an additive map d : R i! R such that d(xy) = d(x)y + xd(y) for all x,y 2 R (resp. d(x 2 ) = d(x)x + xd(x) for all x 2 R). Obviously, every derivation is a Jordan derivation. The converse is in general not true. Herstein (6) proved that the converse is true on a 2-torsion free prime ring. Subsequently, Cusack and Bresar independently extended this re- sult to the case of a 2-torsion free semiprime ring in (4) and (2), respectively. The following two definitions are corresponding to derivations and Jordan deriva- tions, respectively. One is a common extension of derivation, and another is a generalization of Jordan derivation. An additive map µ : R i! R is called a generalized derivation of R if there exists a derivation d of R such that µ(xy) = µ(x)y + xd(y) for all x,y 2 R. d is called an associated derivation of µ. An additive map µ : R i! R is called a generalized Jordan derivation of R if there exists a Jordan derivation d of R such that µ(x 2 ) = µ(x)x + xd(x) for all x 2 R. The map d is called an associated Jordan derivation of µ. When R is a 2-torsion free ring, this definition is equivalent to saying that there exists