A note on multiplicative (generalized) (α,β)-derivations in prime rings - ur Rehman, Nadeem; Al-omary, Radwan M.; Muthana, Najat Mohammed (Annales Mathematicae Silesianae, 2019)
Let R be a prime ring with center Z(R). A map G:R→R is called a multiplicative (generalized) (α,β)-derivation if G(xy) = G(x)α(y)+β(x)g(y) is fulfilled for all x,y∈R, where g:R→R is any map (not necessarily derivation) and α,β:R→R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α,β)-derivations associated with the mappings g and h, respectively, on R and α,β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ◦ H(x) and (v) G(xy) = [α(y),H(x)] for all x,y in an appropriate subset of R.