Amitsur定理能走多远？
How far can we go with Amitsur's theorem?
摘要：
S.A.的一个著名定理。Amitsur证明了多项式环R[x]的Jacobson根对于R的某个诣零理想I等于I[x].本文证明了微分多项式环的情形并非如此，证明了存在一个非诣零的环R和R上的一个导子D使得微分多项式环R[x; D]是Jacobson根.另一方面，我们还证明了微分多项式环R[x; D]的Amitsur定理成立，只要D是局部幂零导子，R是特征p>0的域上的代数.
英文摘要：
A well-known theorem by S.A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x; D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x; D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p>0.
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