Some results on ideals of semiprime rings with multiplicative generalized derivations

ABSTRACTLet R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) F([u,v])=±um[u,v]un, ii) F(u∘v)=±um(u∘v)un, iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.