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Generalized derivations in rings. (English) Zbl 0899.16018
Let $$R$$ be a ring. An additive map $$f\colon R\to R$$ is said to be a generalized derivation if there is a derivation $$d$$ of $$R$$ such that $$f(xy)=f(x)y+xd(y)$$ for all $$x,y\in R$$. The simplest example is a map of the form $$x\mapsto ax+xb$$ where $$a,b$$ are fixed elements in $$R$$; such generalized derivations are called inner. (Inner) generalized derivations have been primarily studied in operator theory, and the intention of the paper under review is to initiate the algebraic study of this concept.
The paper extends several results on derivations of prime rings to generalized derivations. For instance, generalized derivations whose product is again a generalized derivation are characterized, generalized derivations $$f_1$$, $$f_2$$ satisfying $$[f_1(x),f_2(x)]=0$$ for all $$x\in R$$ are considered, and generalized derivations with nilpotent values are treated. The methods are somewhat different from those usually used in the theory of derivations, and are based on a result in a reviewer’s paper [J. Algebra 172, No. 3, 690-720 (1995; Zbl 0827.16013)].

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings
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