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Keywords:
Prime ring; semiprime ring; Banach algebra; Jordan derivation; $(\phi, \varphi )$-derivation
Prime ring; semiprime ring; Banach algebra; Jordan derivation; $(\phi, \varphi )$-derivation
Summary:
Let $\mathcal{R} $ be a semiprime ring with unity $e$ and $\phi $, $\varphi $ be automorphisms of $\mathcal{R} $. In this paper it is shown that if $\mathcal{R} $ satisfies $$2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})$$ for all $x\in \mathcal{R} $ and some fixed integer $n\geq 2$, then $\mathcal{D} $ is an ($\phi $, $\varphi $)-derivation. Moreover, this result makes it possible to prove that if $\mathcal { R}$ admits an additive mappings $\mathcal{D} ,\mathcal{G} \colon \mathcal{R} \rightarrow \mathcal{R} $ satisfying the relations \begin {gather*}\nonumber 2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{G} (x)+\mathcal{G} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{G} (x^{n-1})\,, \\ 2\mathcal{G} (x^n) = \mathcal{G} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})\,, \end {gather*} for all $x\in \mathcal{R} $ and some fixed integer $n\geq 2$, then $\mathcal{D} $ and $\mathcal{G} $ are ($\phi $, $\varphi $)\HH derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.
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