Characterizing Jordan derivations of matrix rings through zero products

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Let be the ring of all n × n matrices over a unital ring R, let M be a 2-torsion free unital -bimodule and let D:→ M be an additive map. We prove that if D()+ D()+D()+ D()=0 whenever,∈ are such that ==0, then D()=δ()+ D(1), where δ:→ 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()+ D()+D()+ D()- D(1)- D(1)=0 whenever ==0. We apply this results to provide that any (generalized) Jordan derivation from into a 2-torsion free -bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if φ:→ is an additive map satisfying φ(+)=φ()+φ() (, ∈), then φ()=φ(1) for all ∈, where φ(1) lies in the centre of. By applying this result we obtain that every Jordan derivation of the trivial extension of by is a derivation.