Semiprime Near-Rings with Multiplicative Generalized (θ, θ)–Derivations

Abstract

Abstract Let N be a semiprime right near-ring and I a semigroup ideal of N . A map f : N → N is called a multiplicative generalized ( θ , θ )–derivation if there exists a multiplicative ( θ , θ )–derivation d : R → R such that f ( xy ) = f ( x ) θ ( y ) + θ ( x ) d ( y ), for all x , y ∈ R . The purpose of this paper is to investigate the following: ( i ) f ( uv ) = ± uv , ( ii ) f ( uv ) = ± vu , ( iii ) f ( u ) f ( v ) = ± uv , ( iv ) f ( u ) f ( v ) = ± vu , ( v ) d ( u ) d ( v ) = θ ([ u , v ]), ( vi ) d ( u ) d ( v ) = θ ( uov ), ( vii ) d ( u ) θ ( v ) = θ ( u ) d ( v ).