AUTOMORPHISMS AND DERIVATIONS ON THE CENTER OF A RING

Abstract

Let $\mathrm{R}$ be a ring, ${\sigma }_1$ an automorphism of $\mathrm{R}$ and ${\delta }_1$ a ${\sigma }_1$-derivation of $\mathrm{R}$. Let ${\sigma }_2$ be an automorphism of $O_1\left(R\right)=R[x;{\sigma }_{1,}{\delta }_1]$, and ${\delta }_2$ be a ${\sigma }_2$-derivation of $O_1\left(R\right)$. Let $S\subseteq Z\left(O_1\right(R\left)\right)$, the center of $O_1\left(R\right)$. Then it is proved that ${\sigma }_i$ is identity when restricted to $S$, and ${\delta }_i$ is zero when restricted to $S$; $i=1,2$. The result is proved for iterated extensions also.