| Summary: | 博士 === 國立臺灣大學 === 數學系研究所 === 86 === Let $R$ be a prime ring with center Z, extended centroid $C$, $Q$ its two-side
dMartindale quotient ring, $\rho$ a nonzero right ideal of $R$ and $\delta , d
$two nonzero derivations of $R$. In the first two sections we study the struc
tures of $R,\delta$ and $d$ when $\delta d(x)$ is central for all$x\in [\rho, rho]$. In $\S$ 3 we extend these results to its polynomial form.More precisely
, we prove the main theorem: Let $f(X_1,\ldots,X_t)$ be anonzero polynomial ov
er $C$. Suppose that $\delta d(f(x_1,\ldots,x_t))\in c$for all $x_1,\ldots,x_t
\in\rho$. Then either char $R=2,\delta =\alpha d$ forsome $\alpha \in C$ and $
d^2=0$, or there exist $p,q\in Q$ such that $\delta=$ ad$(q)$, $d=$ ad$(p)$ wi
th $p\rho=0=q\rho$ and $pq=0$, or $\rho C=eRC$ for some idempotent $e$ in the
socle of $RC$ such that either$f(X_1,\ldots,X_t)$ is central-valued on $eRCe$
or char $R=2$ anddim$_CeRCe=4$.In $\$ 4$ we study the problem concerning annih
ilators of power values ofderivations in prime ring. The follwing main theorem
establishes a unifiedversion of several earlier results in the literature:Sup
pose that $ad([x,y])^n\in Z (d([x,y])^na\in Z$) for all $x,y\in\rho$, where$a in R$ and $n$ is a fixed positive integer. If $[\rho,\rho]\rho]\ne0$ anddim$_C
RC > 4$, then either $ad9\rho)=0$ ($a=0$ resp.) or $d=$ ad$(p)$ forsome $p\in
Q$ such that $p\rho=0$.
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