# On generalized derivations in semiprime rings involving anticommutator

Indexed on: 24 Jan '19Published on: 23 Jan '19Published in: Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

#### Abstract

Let $$\mathfrak {R}$$ be a prime ring with characteristic different from 2 and m, n, k be fixed positive integers. In this paper we study the case when $$\mathfrak {R}$$ admits a generalized derivation $$\mathscr {F}$$ with associated derivation $$\mathfrak {D}$$ such that $$(i)~(\mathscr {F}(x)\circ \mathscr {F}(y))^k=\mathscr {F}(x\circ _ky)~(ii)~ \mathscr {F}(x)\circ _m \mathscr {F}(y) =(\mathscr {F}(x\circ y))^n~(iii)~(\mathscr {F}\left( x\right) \circ \mathscr {F}\left( y\right) )^m =(\mathscr {F}(x\circ y))^n,$$ for all $$x,y \in \mathscr {I}$$, where $$\mathscr {I}$$ is a non-zero ideal of $$\mathfrak {R}$$. Moreover, we also examine the case when $$\mathfrak {R}$$ is a semiprime ring.