Commutativity of semi-derivative prime rings

Abstract

Let $f:R\rightarrow R$ be an additive map** and g be a function of R. If $f\left( xy \right) {=}f\left( x \right)$ $y{+}g\left( x \right) f\left( y \right) {=}f\left( x \right) g\left( y \right) {+}xf\left( y \right)$ and $fg\left( x \right) {=}gf\left( x \right)$ for all $x,y{\in }R$ then f is called a semi-derivation associated with g. Let R be a prime ring with characteristic different from two and $\lambda , \mu {,\, }\sigma {,\, }\tau$ automorphisms of R. Let b be a nonzero element of R and $I,J {,\, }U$ be nonzero ideals of R such that $g{(}I{)\ne 0}$. If one of the following conditions holds then R is commutative: $f{(}I{)\subset }C_{\lambda {,}\mu }\left( J \right) , \quad bf\left( R \right) {\subset }C_{\lambda {,}\mu }\left( R \right) , \quad {\, }f\left[ I{,}J \right] _{\lambda {,}\mu }{=0}, \quad \left[ f\left( I \right) {,}J \right] _{\sigma {,\, }\tau }{\subset }C_{\lambda {,}\mu })$ $\left( U \right) , \quad \left[ f\left( x \right) {,}x \right] _{{1,}\mu }{=0}$ or $\left[ f\left( x \right) {,}g{(}x{)} \right] _{\lambda {,1}}{=0}$, for all $\, x\epsilon I,\, \left[ f\left( I \right) {,}f\left( R \right) \right] _{\lambda {,}\mu }{=0}$. Moreover, we proved that $\left[ f\left( I \right) {,}a \right] _{\sigma {,\, }\tau }{=0}$ if and only if ${f\left[ I{,}a \right] }_{\sigma {,\, }\tau }{=0}$.