Abstract
Motivated by the generalized polynomial identities in [1] and [5], our aim is to generalize some of the results in these works. Precisely, we extend the result in [1] on commuting values of the same generalized derivations to the different generalized derivations case by a short proof. Also as an application, we extend a result in [5] on images of a linear map with derivations to generalized derivations case.
Highlights
We extend a result in [5] on images of a linear map with derivations to generalized derivations case
An additive mapping G : R ! R is called a generalized derivation of R if there exists a derivation of R such that G(xy) = G(x)y + x (y) for all x; y 2 R
For a; b 2 R, it is easy to see that the mapping ax xb is a generalized derivation of R known as inner generalized derivation
Summary
R is called a generalized derivation of R if there exists a derivation of R such that G(xy) = G(x)y + x (y) for all x; y 2 R. Most authors study some generalized polynomial identities on a prime ring and characterize the structure of maps involving in these identities
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