Commutativity of multiplicative $b$-generalized derivations of prime rings
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Consider $\mathscr{R}$ to be an associative prime ring and $\mathscr{K}$ to be a nonzero dense ideal of $\mathscr{R}$. A mapping (need not be additive) $\mathscr{F} : \mathscr{R} \rightarrow \mathscr{Q} _{mr}$ associated with derivation $d : \mathscr{R} \rightarrow \mathscr{R}$ is called a multiplicative $b$-generalized derivation if $\mathscr{F} (\alpha \delta ) = \mathscr{F} (\alpha )\delta + b\alpha d(\delta )$ holds for all $\alpha ,\delta \in \mathscr{R}$ and for any fixed $(0 \neq) b \in \mathscr{Q}_s \subseteq \mathscr{Q}_{mr}$. In this manuscript, we study the commutativity of prime rings when the map $b$-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative $b$-generalized derivation, which improves many results in the literature.
Abstract
Consider $\mathscr{R}$ to be an associative prime ring and $\mathscr{K}$ to be a nonzero dense ideal of $\mathscr{R}$. A mapping (need not be additive) $\mathscr{F} : \mathscr{R} \rightarrow \mathscr{Q} _{mr}$ associated with derivation $d : \mathscr{R} \rightarrow \mathscr{R}$ is called a multiplicative $b$-generalized derivation if $\mathscr{F} (\alpha \delta ) = \mathscr{F} (\alpha )\delta + b\alpha d(\delta )$ holds for all $\alpha ,\delta \in \mathscr{R}$ and for any fixed $(0 \neq) b \in \mathscr{Q}_s \subseteq \mathscr{Q}_{mr}$. In this manuscript, we study the commutativity of prime rings when the map $b$-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative $b$-generalized derivation, which improves many results in the literature.
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