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

Given a unital commutative ring $\mathcal{R}$, $(\mathcal{A},\mathcal{B})$ and $(\mathcal{B},\mathcal{A})$ are bimodules of $\mathcal{M}$ and $\mathcal{N}$, respectively, where $\mathcal{A},\mathcal{B}$ are unitals $\mathcal{R}-$algebras. The $\mathcal{R}-$algebra $\mathcal{G}=$ $\mathcal{G}(\mathcal{A},\mathcal{M},\mathcal{N},\mathcal{B})$ is a generalized matrix algebra described by the Morita context $(\mathcal{A},\mathcal{B},\mathcal{M},\mathcal{N},{\zeta }_{\mathcal{M}\mathcal{N}},{\chi }_{\mathcal{N}\mathcal{M}})$. 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.