The purpose of this paper is to present a brief proof of the well known result of Herstein which states that any Jordan derivation on a prime ring with characteristic not two is a derivation.Throughout this paper all rings will be associative. We shall denote by Z(R) the centre of a ring R. An additive mapping D: R -> R will be called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x,y £ R. We call an additive mappingObviously, every derivation is a Jordan derivation. The converse is in general not true. In this paper we present an alternative proof of the following theorem. for all a , 6 , c € R.(1) is immediate. The proof of (2) holds. All is prepared for the proof of the result below.

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Abstract. The main result: Let R be a 2-torsion free semiprime ring with extended centroid C and let T : R → R be an additive mapping. Suppose that 3T (xyx) = T (x)yx + xT (y)x + xyT (x) holds for all x, y ∈ R. Then there exists an element λ ∈ C such that T (x) = λx for all x ∈ R.This research has been motivated by the work of Brešar [4] and Zalar [8]. Throughout, R will represent an associative ring with center Z (R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. As usual the commutator xy − yx will be denoted by [x, y] . We shall use basis commutator identities [xy, z] Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x 2 ) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = [a, x] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [6] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation (see [2] for an alternative proof). Cusack [5] generalized Herstein's result on 2-torsion free semiprime rings (see [3] for an alternative proof). We follow Zalar [8] and call an additive mapping T : R → R a left (right) centralizer in case T (xy) = T (x)y (T (xy) = xT (y)) holds for all x, y ∈ R. This concept appears naturally in C * -algebras. In ring theory it is more common to work with module homomorphisms. Ring theorists would write that T : R R → R R is 2000 Mathematics Subject Classification. 16N60, 39B05.

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Summary. Let R be a ring. A bi-additive symmetric mapping D(., .): R x R ~R is called a symmetric bi-derivation if, for any fixed y e R, a mapping x ~ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivation D(., .): R x R ~R, where R is a prime ring of characteristic not two, with the property D(x, x)x = xD(x, x), x e R, forces R to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, if R is a prime ring of characteristic not two and D I, D 2 are nonzero derivations on R, then the mapping x ~,DI(D2(x)) cannot be a derivation, is also presented. PreliminariesT h r o u g h o u t this paper all rings will be associative. W e shall denote by Z(R) the center of a ring R. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semi-prime if aRa = (0) implies a = 0. We shall write [x, y] for xy -yx and use the identities