Characterizing Jordan derivations of matrix rings through zero products

Abstract

Let \Mn be the ring of all $n \times n$ matrices over a unital ring $\mathcal{R}$, let $\mathcal{M}$ be a 2-torsion free unital \Mn-bimodule and let D:\Mn\rightarrow \mathcal{M} be an additive map. We prove that if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B\in \Mn are such that \A\B=\B\A=0, then D(\A)=\delta(\A)+\A D(\textbf{1}), where \delta:\Mn\rightarrow \mathcal{M} is a derivation and $D(\textbf{1})$ lies in the centre of $\mathcal{M}$. It is also shown that $D$ is a generalized derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A D(\textbf{1})\B-\B D(\textbf{1})\A=0 whenever \A\B=\B\A=0. We apply this results to provide that any (generalized) Jordan derivation from \Mn into a 2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if \varphi:\Mn\rightarrow \Mn is an additive map satisfying \varphi(\A \B+\B \A)=\A\varphi(\B)+\varphi(\B)\A \quad (\A,\B \in \Mn), then \varphi(\A)=\A\varphi(\textbf{1}) for all \A\in \Mn, where $\varphi(\textbf{1})$ lies in the centre of \Mn. By applying this result we obtain that every Jordan derivation of the trivial extension of \Mn by \Mn is a derivation.Comment: To appear in Mathematica Slovac