Representational repleteness of differential operator rings
type: Item
instance of: scholarly article
title: Representational repleteness of differential operator rings
zbMATH Open document ID: 0873.16014
author: Gordan MacLaren Low
published in: Journal of Algebra
publication date: 1997-10-27T00:00:00Z
review text: Let \(P\) be a prime ideal of Noetherian ring \(R\), and let \(E\) be an indecomposable injective right \(R\)-module with assassinator \(P\). Then \(P\) is said to be (right) representationally replete (rep. rep.) if every prime ideal which could be involved in \(E\) is involved, in the following sense: Let \(A_n(P)\) be the set of associated primes of the \(n\)th layer of \(E\) and let \(X_n(P)\) be the set of primes which are \((n-1)\) second layer links away from \(P\) in the graph of links. Then \(P\) is right rep. rep. if \(\bigcup^\infty_{n=1}A_n(P)=\bigcup^\infty_{n=1}X_n(P)\); it is right strongly rep. rep. if \(A_n(p)=X_n(p)\) for all \(n\). (These concepts are independent of the choice of \(E\), as shown in [\textit{K. A. Brown} and \textit{R. B. Warfield} jun., J. Algebra 116, 294-315 (1988; Zbl 0652.16006)], where the terms used above are also defined.) In the paper under review, which is a sequel to [\textit{G. M. Low}, Injective modules and representational repleteness, J. Algebra (to appear)], the case where \(R\) is a ring of differential operators is examined in detail. A sample result is the following. Let \(R=S[\theta_1,\dots,\theta_n;\delta_1,\dots,\delta_n]\) be a differential operator ring, where \(S\) is a commutative Noetherian \(\mathbb{Q}\)-algebra and \(\delta_1,\dots,\delta_n\) are commuting \(\mathbb{Q}\)-derivations of \(S\). Let \(P\) be a prime ideal of \(R\), and set \(M=P\cap S\). If \(S_M\) is a regular local ring then \(P\) is rep. rep.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S32
zbMATH DE Number: 954741
zbMATH Keywords: indecomposable injective right modules
zbMATH Keywords: prime ideals of Noetherian rings
zbMATH Keywords: assassinators
zbMATH Keywords: second layer links
zbMATH Keywords: representational repleteness
zbMATH Keywords: rings of differential operators
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: jabr.1996.0361
OpenAlex ID: W2051097023
DOI: 10.1006/JABR.1996.0361
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description: scientific article; zbMATH DE number 954741