A generalization of multiplicative (generalized)-derivatons

Abstract

Let R be a semiprime ring and L be a semigroup ideal of R. The main object in this paper is to study the following situations in semiprime rings: When F is a multiplicative (α,1)-(generalized) derivation associated with a map d, (i) F(xy)±α(x)α(y)=0 for x,y∈L. (ii) F(x)F(y)±α(x)α(y)=0 for all x,y∈L. When F is a multiplicative (1,α)-(generalized) derivation associated with a map d, (iii) F(xy)±xy=0 for all x,y∈L. (iv) F(x)F(y)±xy=0 for all x,y∈L.