On generalized Jordan derivations of generalized matrix algebras
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Let $\mathfrak{R}$ be a commutative ring with unity, $\mathrm{A}$ and $\mathrm{B}$ be $\mathfrak{R}$-algebras, $\mathrm{M}$ be a $(\mathrm{A}, \mathrm{B})$-bimodule and $\mathrm{N}$ be a $(\mathrm{B}, \mathrm{A})$-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.
Abstract
Let $\mathfrak{R}$ be a commutative ring with unity, $\mathrm{A}$ and $\mathrm{B}$ be $\mathfrak{R}$-algebras, $\mathrm{M}$ be a $(\mathrm{A}, \mathrm{B})$-bimodule and $\mathrm{N}$ be a $(\mathrm{B}, \mathrm{A})$-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.
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