Commutativity of semi-derivative prime rings

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