On Generalized Jordan derivations of generalized matrix algebras

Commun. Korean Math. Soc. Published online May 7, 2020

Mohammad Ashraf and Aisha Jabeen
Aligarh Muslim University, Jamia Millia Islamia

Abstract : Let $\mathfrak{R}$ be a commutative ring with unity, $\mathrm{A}~\& ~\mathrm{B}$ be $\mathfrak{R}$-algebra, $\mathrm{M}$ be $(\mathrm{A}, \mathrm{B})$-bimodule and $\mathrm{N}$ be $(\mathrm{B}, \mathrm{A})$-bimodule. The $\mathfrak{R}$-algebra $\mathfrak{S}=\mathfrak{S}(\mathrm{A}, \mathrm{M}, \mathrm{N}, \mathrm{B})$ is a generalized matrix algebra defined by the Morita context $(\mathrm{A}, \mathrm{B}, \mathrm{M}, \mathrm{N}, \xi_{\mathrm{M}\mathrm{N}},
\Omega_{\mathrm{N}\mathrm{M}}).$ In this article, we study generalized derivation and generalized Jordan derivation on generalized matrix algebras and prove that every generalized Jordan derivation can be written as the sum of a generalized derivation and antiderivation with some limitations. Also, we show that every generalized Jordan derivation is a generalized derivation on trivial generalized matrix algebra over a field.

Keywords : generalized matrix algebras, generalized derivation, Jordan derivation