ON (m,n)–JORDAN CENTRALIZERS IN RINGS AND ALGEBRAS

Let m ≥ 0, n ≥ 0 be fixed integers with m + n 6= 0 and let R be a ring. It is our aim in this paper to investigate additive mapping T : R → R satisfying the relation (m + n)T (x2) = mT (x)x + nxT (x) for all x ∈ R. This research is a continuation of our earlier work ([11]). 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 define [y, x]n inductively as follows: [y, x]1 = [y, x] , [y, x]n+1 = [[y, x]n , x] . We shall use the commutator identities [xy, z] = [x, z] y+x [y, z] and [x, yz] = [x, y] z+y [x, z] , for all x, y, z ∈ R. A mapping F, which maps a ring R into itself, is called commuting on R in case [F (x), x] = 0 holds for all x ∈ R. 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) = 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 ([8]) asserts that any Jordan derivation on a prime ring with char(R) 6= 2 is a derivation. A brief proof of Herstein’s result can be found 2010 Mathematics Subject Classification. 16W10, 46K15, 39B05.