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Abstract. The purpose of this paper is to prove the following result. Let m ≥ 1, n ≥ 1 be some fixed integers and let R be a prime ring with char(R) = 0 or (m + n) 2 < char(R). Suppose there exists an additive mapping T : R → R satisfying the relation 2(m + n) 2 T (x 3 ) = m(2m + n)T (x)x 2 + 2mnxT (x)x + n(2n + m)x 2 T (x) for all x ∈ R. In this case T is a two-sided centralizer.Throughout, R will represent an associative ring with center Z(R). Given an integer n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 implies x = 0. As usual the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [xy, z] Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0 and is semiprime in case aRa = (0) implies a = 0. We denote by char(R) the characteristic of a prime ring R. An additive mapping D : R → R, where R is an arbitrary ring, 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 ([10]) asserts that any Jordan derivation on a prime ring with char(R) = 2 is a derivation. A brief proof of Herstein's result can be found in [3]. Cusack 2010 Mathematics Subject Classification. 16W10, 46K15, 39B05.

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Abstract. A classical result of Herstein asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein's theorem. Let R be a prime ring with char(R) = 0 or 4 < char(R), and let D : R → R be an additive mapping satisfying either the relation D(This research has been motivated by the recent work of Vukman ([14]). Throughout, R will represent an associative ring with center Z(R). As usual we write [x, y] for xy − yx. Given an integer n ≥ 2, a ring R is said to be ntorsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0 and is semiprime in case aRa = (0) implies a = 0. An additive mapping D : R → R, where R is an arbitrary ring, 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) = [x, a] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein ([11]) asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of Herstein's result can be found in [7]. Cusack ([10]) generalized Herstein's result to 2-torsion free semiprime rings (see also [3] for an alternative proof). Let us point out that Beidar, Brešar, Chebotar and Martindale ([1]) have considerably generalized Herstein's theorem. A generalization of Herstein's theorem can be found also in [8].