On Pair of Generalized Derivations in Rings

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 $R$.