Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R.

Abstract. The concept of derivations as well as generalized derivations (i.e.

Abstract. Let R be a 2-torsion free prime ring, and let J be a nonzero Jordan ideal and a subring of R. In the present paper it is shown that if d is an additive mapping of IntroductionThroughout this paper R will denote an associative ring with center Z(R). Recall that R is prime if aRb -(0) implies that a = 0 or b = 0. As usual [x, y] and x o y will denote the commutator xy -yx and anticommutator xy + yx, respectively. A ring R is said to be 2-torsion free, if whenever 2x = 0, with x G R, then x = 0. An additive subgroup J of R is said to be a Jordan ideal of R if u o r € J, for all u G J, r € R. An additive mapping d : R -• R is called a derivation if d(xy) = d(x)y + xd(y), holds for all pairs x, y G R. An additive mapping d : R -> R is called a Jordan derivation if d(x 2 ) = d(x)x + xd(x)holds for all x G R. Obviously, every derivation on a ring R is a Jordan derivation. The converse is, in general, not true. A well known result due to Herstein [6] shows that every Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result is presented in [4]. This result was generalized by many authors (cf. [5]). In the present paper, our objective is to generalize this result for derivations defined on a subset of a prime rings. Preliminary resultsTo facilitate our discussion, we introduce abbreviation ¿(x, y) = d(xy) -d(x)y -xd(y). We shall make use of commutator identities:

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Abstract. A semiderivation of a ring R is an additive mapping / : R -• R together with a function g : R -• R such that f(xy) = f{x)g(y) + xf(y) = f(x)y + g(x)f(y) andIf / is a non-zero semiderivation of a prime ring R, then it is well known that g must necessarily be an endomorphism. Let R be a prime ring with center Z(R), f a non-zero semiderivation with associated endomorphism g which is one-one & onto, and a, r be two automorphisms of R such that fa = erf, fr = rf, go = ag, gr = rg. Suppose that U is a non-zero (cr, r)-Lie ideal of R and C(R)a,T = {c G R \ ca(x) = T(X)C,for all x € R}. In the present paper it is shown that (i) if char R IntroductionLet R be a ring with center Z(R), and U an additive subgroup of R.