On certain functional equation arising from (m,n)- Jordan centralizers in prime rings

Abstract

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.