Generalized Jordan derivations on prime rings and standard operator algebras. (English) Zbl 1058.16031

Jing, Wu; Lu, Shijie

Taiwanese J. Math. 7, No. 4, 605-613 (2003).

Let \(R\) be a ring. An additive map \(\delta\colon R\to R\) is called a generalized derivation if there exists a derivation \(\tau\colon R\to R\) such that \(\delta(xy)=\delta(x)y+x\tau(y)\) for all \(x,y\in R\). The authors define a generalized Jordan derivation as an additive map \(\delta\colon R\to R\) such that there exists a Jordan derivation \(\tau\colon R\to R\) satisfying \(\delta(x^2)=\delta(x)x+x\tau(x)\) for all \(x\in R\). In a similar fashion they define a generalized Jordan triple derivation, extending the known concept of a Jordan triple derivation. Using a standard approach they extend known results on Jordan (triple) derivations on prime rings and standard operator algebras to the generalized ones.

Reviewer: M. Brešar (Maribor)

MSC:

16W10	Rings with involution; Lie, Jordan and other nonassociative structures
16N60	Prime and semiprime associative rings
16W20	Automorphisms and endomorphisms
47B47	Commutators, derivations, elementary operators, etc.
47L10	Algebras of operators on Banach spaces and other topological linear spaces

Keywords:

generalized Jordan derivations; prime rings; standard operator algebras; additive maps; generalized derivations