A note on multiplicative (generalized)-derivations and left ideals in semiprime rings

Abstract

Let R be a semiprime ring and $$\lambda$$ be a nonzero left ideal of R. A mapping $$d : R \rightarrow R$$ (not necessarily a derivation nor an additive map) which satisfies $$d(xy) = d(x)y+xd(y)$$ for all $$x, y \in R$$ is called a multiplicative derivation of R. A mapping $$F:R\rightarrow R$$ (not necessarily additive) is called a multiplicative (generalized)-derivation of R if there exists a map $$d:R \rightarrow R$$ (not necessarily a derivation nor an additive map) such that $$F(xy)=F(x)y+xd(y)$$ holds for all $$x,y\in R$$. The objective of this paper is to study the following identities: (1) $$[d(x),F(y)]=\pm [x,y]$$, (2) $$[d(x),F(y)]=\pm x\circ y$$, (3) $$[d(x),F(y)]=0$$, (4) $$F([x,y])\pm [\delta (x),\delta (y)]\pm [x,y]=0$$, (5) $$d'([x,y])\pm [\delta (x),\delta (y)]\pm [x,y]=0$$, (6) $$d'([x,y])\pm [\delta (x),\delta (y)]=0$$, (7) $$F(x \circ y)\pm \delta (x)\circ \delta (y) \pm x\circ y=0$$, (8) $$d'(x \circ y)\pm \delta (x)\circ \delta (y) \pm x\circ y=0$$, (9) $$d'(x \circ y)\pm \delta (x)\circ \delta (y)=0$$, for all $$x,y\in \lambda$$, where F is a multiplicative (generalized)-derivation of R associated to the map d, and $$d'$$, $$\delta$$ are multiplicative derivations of R.