On Pair of Generalized Derivations in Rings

Document Type : Original Article

Authors

Asma Ali

Md Hamidur Rahaman

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India.

10.22034/kjm.2019.97174

Abstract

Let $R$ be an associative ring with extended centroid $C$, let $G$ and $F$ be generalized derivations of $R$ associated with nonzero derivations $\delta$ and $d$, respectively, and let $m, k, n \geq1$ be fixed integers. In the present paper, we study the situations: (i)$F(x)\circ_{m}G(y)=(x \circ_{n} y)^{k}$, (ii) $[F(x),y]_{m}+[x,d(y)]_{n}=0$ for all $y, x$ in some appropriate subset of $R$.

Keywords

Prime rings

semiprime rings

generalized derivations

extended centroid

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