, Volume 86, Issue 1-2, pp 65-79
Date: 29 May 2013

On multiplicative (generalized)-derivations in prime and semiprime rings

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Abstract

Let R be a ring. A map \({F : R \rightarrow R}\) is called a multiplicative (generalized)-derivation if F(xy) = F(x)yxg(y) is fulfilled for all \({x, y \in R}\) where \({g : R \rightarrow R}\) is any map (not necessarily derivation). The main objective of the present paper is to study the following situations: (i) \({F(xy) \pm xy \in Z}\) , (ii) \({F(xy) \pm yx \in Z}\) , (iii) \({F(x)F(y) \pm xy \in Z}\) and (iv) \({F(x)F(y) \pm yx \in Z}\) for all x, y in some appropriate subset of R. Moreover, some examples are also given.

This research is partially supported by the research grants from UGC, India (Grant No. F. PSW-099/10-11 and 39-37/2010(SR), respectively).