Rings of differential operators and zero divisors
This paper is concerned with the structure of the ring \({\mathcal D}(A)\) of all k-linear differential operators on a commutative affine k-algebra A, where k is an algebraically closed field of characteristic zero. It is not assumed that A is reduced, nor is any restriction placed on K.dim(A). The author proves that \({\mathcal D}(A)\) is semiprime Goldie if and only if A has an artinian quotient ring. He also develops an explicit description of the prime radical N of \({\mathcal D}(A)\), from which it follows that \(N^{n+1}=0\) where \(n=K.\dim (A)\). Finally, an analogue of Nakai's conjecture is obtained: if \({\mathcal D}(A)\) is generated by operators of order at most 1, then A is reduced.
scientific article; zbMATH DE number 4112798