Lie (Jordan) $ \sigma- $centralizer at the zero products on generalized matrix algebra

Abstract

<p>Given a unital commutative ring $ \mathscr{R} $, $ (\mathscr{A}, \mathscr{B}) $ and $ (\mathscr{B}, \mathscr{A}) $ are bimodules of $ \mathscr{M} $ and $ \mathscr{N} $, respectively, where $ \mathscr{A}, \mathscr{B} $ are unitals $ \mathscr{R}- $algebras. The $ \mathscr{R}- $algebra $ \mathscr{G} = $ $ \mathscr{G}(\mathscr{A}, \mathscr{M}, \mathscr{N}, \mathscr{B}) $ is a generalized matrix algebra described by the Morita context $ (\mathscr{A}, \mathscr{B}, \mathscr{M}, \mathscr{N}, \zeta_{\mathscr{M}\mathscr{N}}, \chi_{\mathscr{N}\mathscr{M}}) $. The present study investigated the structure of Lie (Jordan) $ \sigma- $centralizers at the zero products on order two generalized matrix algebra and established that each Jordan $ \sigma- $centralizer at the zero products is a $ \sigma- $centralizer at the zero product on order two generalized matrix algebra. We also provided sufficient and necessary conditions under which a Lie $ \sigma- $centralizer at the zero product is proper on an order two generalized matrix algebra.</p>