Rings of differential operators and zero divisors
type: Item
instance of: scholarly article
title: Rings of differential operators and zero divisors
zbMATH Open document ID: 0679.16002
DOI: 10.1016/0021-8693(89)90178-6
author: Ian M. Musson
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 13B10
zbMATH DE Number: 4112798
zbMATH Keywords: differential operators
zbMATH Keywords: affine k-algebra
zbMATH Keywords: semiprime Goldie
zbMATH Keywords: artinian quotient ring
zbMATH Keywords: prime radical
zbMATH Keywords: Nakai's conjecture
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90178-6
OpenAlex ID: W2085107948
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description: scientific article; zbMATH DE number 4112798