Additive mappings act as a generalized left $$(\alpha , \beta )$$ ( α , β ) -derivation in rings

Abstract

If $$F,D:R\rightarrow R$$ are additive mappings satisfying $$F(x^{n}y^{n})=\alpha (x^n)F(y^{n})+\beta (y^n)D(x^{n})$$ for all $$x,y\in R$$ , where $$\alpha ,\beta $$ are automorphisms on R, then F becomes generalized left $$(\alpha , \beta )$$ -derivation with associated Jordan left $$(\alpha , \beta )$$ -derivation D on R under suitable torsion restriction.