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 $D(\A)=δ(\A)+\A D(\textbf{1})$, where $δ:\Mn\rightarrow \mathcal{M}$ 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 $φ:\Mn\rightarrow \Mn$ is an additive map satisfying $φ(\A \B+\B \A)=\Aφ(\B)+φ(\B)\A \quad (\A,\B \in \Mn)$, then $φ(\A)=\Aφ(\textbf{1})$ for all $\A\in \Mn$, where $φ(\textbf{1})$ 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.