ON η−GENERALIZED DERIVATIONS IN RINGS WITH JORDAN INVOLUTION
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Let $\Ka$ be a ring. An additive map $\mathfrak{u}^\diamond\rightarrow \mathfrak{u}$ is called Jordan involution on $\Ka$ if $(\mathfrak{u}^\diamond)^\diamond=\mathfrak{u}$ and $(\mathfrak{u}\mathfrak{v}+\mathfrak{v}\mathfrak{u})^\diamond=\mathfrak{u}^{\diamond}\mathfrak{v}^{\diamond}+\mathfrak{v}^{\diamond}\mathfrak{u}^{\diamond}$ for all $\mathfrak{u},\mathfrak{v}\in \Ka$. If $\Theta$ is a (non-zero) $\eta-$generalized derivation on $\Ka$ associated with a derivation $\Omega$ on $\Ka$, then it is shown that $\Theta(\mathfrak{u})=\gamma \mathfrak{u}$ for all $\mathfrak{u}\in \Ka$ such that $\gamma\in \Xi$ and $\gamma^2=1$, whenever $\Theta$ possesses $[\Theta(\mathfrak{u}), \Theta(\mathfrak{u}^\diamond)]=[\mathfrak{u},\mathfrak{u}^\diamond]$ for all $\mathfrak{u}\in \Ka$.
Abstract
Let $\Ka$ be a ring. An additive map $\mathfrak{u}^\diamond\rightarrow \mathfrak{u}$ is called Jordan involution on $\Ka$ if $(\mathfrak{u}^\diamond)^\diamond=\mathfrak{u}$ and $(\mathfrak{u}\mathfrak{v}+\mathfrak{v}\mathfrak{u})^\diamond=\mathfrak{u}^{\diamond}\mathfrak{v}^{\diamond}+\mathfrak{v}^{\diamond}\mathfrak{u}^{\diamond}$ for all $\mathfrak{u},\mathfrak{v}\in \Ka$. If $\Theta$ is a (non-zero) $\eta-$generalized derivation on $\Ka$ associated with a derivation $\Omega$ on $\Ka$, then it is shown that $\Theta(\mathfrak{u})=\gamma \mathfrak{u}$ for all $\mathfrak{u}\in \Ka$ such that $\gamma\in \Xi$ and $\gamma^2=1$, whenever $\Theta$ possesses $[\Theta(\mathfrak{u}), \Theta(\mathfrak{u}^\diamond)]=[\mathfrak{u},\mathfrak{u}^\diamond]$ for all $\mathfrak{u}\in \Ka$.
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