Additive mappings satisfying algebraic conditions in rings

Let R be any (n+1)!-torsion free ring and F,D: R → R be additive mappings satisfying (Formula presented.) for all x∈ R, where n is a fixed integer and α, β are automorphisms of R. Then, D is Jordan left (α, β)-derivation and F is generalized Jordan left (α, β)-derivation on R and if additive mappings F and D satisfying (Formula presented.) for all x∈ R. Then, D is Jordan (α, β)-derivation and F is generalized Jordan (α, β) -derivation on R. At last some immediate consequences of the above theorems have been given. © 2014 Springer-Verlag Italia.