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Let 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) ii) iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I. [ABSTRACT FROM AUTHOR]

Let R be a prime or semiprime ring equipped with an involution * and α be an automorphism of R: An additive mapping T : R → R is called a left (resp. right) α-*centralizer of R if T (xy) = T (x) α (y*) (resp. T (xy) = α (x*) T (y)) holds for all x,y ∈ R; where α is an endomorphism of R. A left (resp. right) Jordan α-*centralizer T : R → R is an additive mapping such that T (x²) = T (x²) α(x*) (resp. T (x²) = α(x*)T (x)) holds for all x ∈ R: In this paper, we obtain some results about Jordan α-*centralizer of R with involution. [ABSTRACT FROM AUTHOR]

We extend some well known commutativity results concerning a nonzero square closed ∗-Lie ideal and generalized α,β-derivations of ∗-prime rings. [ABSTRACT FROM AUTHOR]

Let R be a *-prime ring with characteristic different from two and U ≠ 0 be a square closed *-Lie ideal of R. An additive mapping F : R → R is called an generalized derivation if there exits a derivation d : R → R such that F(xy) = F(x)y+xd(y). In the present paper, it is shown that U ⊆ Z if R is a *-prime ring which admits a generalized derivation satisfying several conditions that are associated with a derivation commuting with *. [ABSTRACT FROM AUTHOR]