Characterizing Jordan derivations of matrix rings through zero products

Rok, strany: 2015, 1277 - 1290

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

ISO 690:

Ghahramani, H. 2015. Characterizing Jordan derivations of matrix rings through zero products. In Mathematica Slovaca, vol. 65, no.6, pp. 1277-1290. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0089

APA:

Ghahramani, H. (2015). Characterizing Jordan derivations of matrix rings through zero products. Mathematica Slovaca, 65(6), 1277-1290. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0089