A note on b-generalized (α,β)-derivations in prime rings

Abstract

Let R be a prime ring, let 0 ≠ b ∈ R {0\neq b\in R} , and let α and β be two automorphisms of R. Suppose that F : R → R {F:R\rightarrow R} , F 1 : R → R {F_{1}:R\rightarrow R} are two b-generalized ( α , β ) {(\alpha,\beta)} -derivations of R associated with the same ( α , β ) {(\alpha,\beta)} -derivation d : R → R d:R\rightarrow R , and let G : R → R G:R\rightarrow R be a b-generalized ( α , β ) (\alpha,\beta) -derivation of R associated with ( α , β ) (\alpha,\beta) -derivation g : R → R g:R\rightarrow R . The main objective of this paper is to investigate the following algebraic identities: (1) F ⁢ ( x ⁢ y ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+\alpha(xy)+\alpha(yx)=0} , (2) F ⁢ ( x ⁢ y ) + G ⁢ ( x ) ⁢ α ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+G(x)\alpha(y)+\alpha(yx)=0} , (3) F ⁢ ( x ⁢ y ) + G ⁢ ( y ⁢ x ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+G(yx)+\alpha(xy)+\alpha(yx)=0} , (4) F ⁢ ( x ) ⁢ F ⁢ ( y ) + G ⁢ ( x ) ⁢ α ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(x)F(y)+G(x)\alpha(y)+\alpha(yx)=0} , (5) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( x ⁢ y ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(xy)=0} , (6) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) = 0 {F(xy)+d(x)F_{1}(y)=0} , (7) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(yx)=0} , (8) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( x ⁢ y ) + α ⁢ ( y ⁢ x ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(xy)+\alpha(yx)=0} , (9) F ⁢ ( x ⁢ y ) + d ⁢ ( x ) ⁢ F 1 ⁢ ( y ) + α ⁢ ( y ⁢ x ) - α ⁢ ( x ⁢ y ) = 0 {F(xy)+d(x)F_{1}(y)+\alpha(yx)-\alpha(xy)=0} , (10) [ F ⁢ ( x ) , x ] α , β = 0 {[F(x),x]_{\alpha,\beta}=0} , (11) ( F ⁢ ( x ) ∘ x ) α , β = 0 {(F(x)\circ x)_{\alpha,\beta}=0} , (12) F ⁢ ( [ x , y ] ) = [ x , y ] α , β {F([x,y])=[x,y]_{\alpha,\beta}} , (13) F ⁢ ( x ∘ y ) = ( x ∘ y ) α , β {F(x\circ y)=(x\circ y)_{\alpha,\beta}} for all x , y {x,y} in some suitable subset of R.