CHARACTERIZING JORDAN DERIVATIONS OF MATRIX RINGS THROUGH ZERO PRODUCTS

Let M-n(R) be the ring of all n x n matrices over a unital ring R, let M be a 2-torsion free unital M-n(R)-bimodule and let D: M-n(R) -> M be an additive map. We prove that if D(a)b + aD(b) + D(b) a + bD(a) = 0 whenever a, b is an element of M-n(R) are such that ab = ba = 0, then D(a) = delta(a) + aD(1), where delta : M-n(R) -> M is a derivation and D(1) lies in the centre of M. It is also shown that D is a generalized derivation if and only if D(a)b + aD(b) + D(b)a + bD(a) - aD(1)b - bD(1)a = 0 whenever ab = ba = 0. We apply this results to provide that any (generalized) Jordan derivation from M-n(R) into a 2-torsion free M-n(R)-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if phi: M-n(R) -> M-n(R) is an additive map satisfying phi(ab + ba) = a phi(b) + phi(b)a (a, b is an element of M-n(R)), then phi(a) = a phi(1) for all a is an element of M-n(R), where phi(1) lies in the centre of M-n(R). By applying this result we obtain that every Jordan derivation of the trivial extension of M-n(R) by M-n(R) is a derivation. (C) 2015 Mathematical Institute Slovak Academy of Sciences