Communications of the
Korean Mathematical Society CKMS

Let $\mathscr{K}$ be a ring. An additive map $\mathfrak{u}^\diamond\rightarrow \mathfrak{u}$ is called Jordan involution on $\mathscr{K}$ 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 $\mathfrak{u},\mathfrak{v}\in \mathscr{K}$. If $\Theta$ is a (non-zero) $\eta-$generalized derivation on $\mathscr{K}$ associated with a derivation $\Omega$ on $\mathscr{K}$, then it is shown that $\Theta(\mathfrak{u})=\gamma \mathfrak{u}$ for all $\mathfrak{u}\in \mathscr{K}$ 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 $\mathfrak{u}\in \mathscr{K}$.

Keywords: Prime ring, Jordan involution, $\eta-$generalized derivation, strong commutativity preserving (SCP) map

MSC numbers: 16N60, 16W10, 16W25