Action of multiplicative (generalized)-derivations and related maps on square closed Lie ideals in prime rings

Abstract

Let $\mathcal{R}$ be a prime ring and $L$ a nonzero square closed Lie ideal of $\mathcal{R}$. Suppose $F,G,H\colon \mathcal{R}\to \mathcal{R}$\break are three multiplicative (generalized)-derivations associated with the maps $\delta,g, h\colon \mathcal{R}\to \mathcal{R}$ respectively which are not necessarily additive or derivations. Assume that $E,T\colon \mathcal{R}\to \mathcal{R}$ be any two maps (not necessarily additive).

Let $d\colon \mathcal{R}\to \mathcal{R}$ be a nonzero derivation of $\mathcal{R}$. In the present article, following identities are studied

 (1) $d(x)F(y)+G(y)d(x)\pm (E(x)y+yT(u))=0,\quad$ $(2)\ H(xy)+G(y)F(x)\pm (E(y)x+xT(y))=0$,

(3) $T(xy)+G(x)y\pm (yx+xy)=0, \quad$  $(4)\ F(x)F(y)+T(x)y\pm yx=0,$ (5) $d(x)d(y)+T(x)y+F(yx)=0$, for all $x,y\in L$.