# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Generalized derivations in rings. (English) Zbl 0899.16018

Let $R$ be a ring. An additive map $f:R\to R$ is said to be a generalized derivation if there is a derivation $d$ of $R$ such that $f\left(xy\right)=f\left(x\right)y+xd\left(y\right)$ for all $x,y\in R$. The simplest example is a map of the form $x↦ax+xb$ 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 $\left[{f}_{1}\left(x\right),{f}_{2}\left(x\right)\right]=0$ 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 (associative rings and algebras) 16N60 Prime and semiprime associative rings