ON (m,n)-JORDAN DERIVATIONS AND COMMUTATIVITY OF PRIME RINGS
The purpose of this paper is to prove the following result. Let m ≥ 1, n ≥ 1 be some fixed integers with m = n, and let R be a prime ring with char(R) = 2mn(m + n) m − n . Suppose there exists a nonzero additive mapping D : R → R satisfying the relation (m + n)D(x 2 ) = 2mD(x)x + 2nxD(x) for all x ∈ R ((m, n)-Jordan derivation). If either char(R) = 0 or char(R) > 3 then D is a derivation and R is commutative. This research is related to our earlier work  and  . Throughout, R will
... out, 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. For x, y ∈ R we write [y, x] 1 = [y, x] = yx − xy, and for n ≥ 1, [y, x] n = [y, x] n−1 , x . 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. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein  has proved 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  . Cusack  has proved Herstenin's theorem for 2-torsion free semiprime rings (see  for an alternative proof). An additive mapping D : R → R is called a left derivation if D(xy) = yD(x) + xD(y) holds for all pairs x, y ∈ R and is called a left Jordan derivation (or Jordan left derivation) in case D(x 2 ) = 2xD(x) is fulfilled for all x ∈ R. The concepts of left derivation and left Jordan derivation were introduced by Brešar and Vukman in  . One can easily prove (see  ) that the existence of a nonzero 2000 Mathematics Subject Classification: 16N60.