Let R be a semiprime ring with center Z(R). A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)derivation if there exists a map f : R → R (not necessarily a derivation nor an additive map) such that F (xy) = F (x)y + xf (y) holds for all x, y ∈ R.
Let R be a prime ring, d, δ two derivations of R, L a noncentral Lie ideal of R and 0 = a ∈ R. The main object in this paper is to discuss the situations a(d(x)x − xδ(x)) n = 0 for all x ∈ L and a(d(x)x − xδ(x)) ∈ Z(R) for all x ∈ L, where n ≥ 1 is a fixed integer.
LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.
The main object in this article is to discuss the generalized derivation acting as homomorphism or anti-homomorphism in some nonzero left ideal of a semiprime ring.
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