Jordan Endo-Biderivations on prime rings

Abstract

For a ring $\mathcal{R}$, set $\mathcal{S}=\mathcal{R}\times\mathcal{R}$. Our goal in this essay is to offer the concepts of Endo-Biderivation, Jordan Endo-Biderivation and Quasi Endo-Biderivation on $\mathcal{S}$: A bi-additive mapping $\xi: \mathcal{S} \rightarrow \mathcal{S}$ is referred to as Endo-Biderivation on $\mathcal{S}$ if $\xi(\alpha\mathfrak{b}, \mathfrak{c}\mathcal{d})= \xi(\alpha, \mathfrak{c}) (\mathfrak{b}, \mathcal{d}) + (\alpha, \mathfrak{c}) \xi(\mathfrak{b}, \mathcal{d})$ fulfilled for any $\alpha, \mathfrak{b}, \mathfrak{c},\mathcal{d}\in \mathcal{R}$. A mapping $\xi$ is referred to as Jordan Endo-Biderivation on $\mathcal{S}$ if $\forall\ \alpha\ , \mathfrak{c} \in \mathcal{R}$, then $\xi(\alpha^2, \mathfrak{c}^2) = \xi(\alpha, \mathfrak{c}) (\alpha, \mathfrak{c}) + ( \alpha, \mathfrak{c})\xi(\alpha , \mathfrak{c})$. It's shown that in this occurrence that $\mathcal{S}$ is prime-ring (P.R) of characteristic $\neq2, \& \ \xi: \mathcal{S} \longrightarrow \mathcal{S}$ is a Jordan Endo-Biderivation, then $\xi$ is Endo-Biderivation.