The influence of ideal structure on representation theory
type: Item
instance of: scholarly article
title: The influence of ideal structure on representation theory
zbMATH Open document ID: 0652.16006
DOI: 10.1016/0021-8693(88)90219-0
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let R be a Noetherian ring. The existence of a uniform right R-module M that is the non-split extension of a uniform submodule U with \(Ann(U)=Ass(U)=P\) and \(U=Ann_ M(P)\) by a uniform module V with \(Ann(V)=Ass(V)=Q\), allows considerable insight into the prime ideal structure of R. Thus, \textit{A. V. Jategaonkar}'s ``Main Lemma'' [Localization in Noetherian Rings (Lond. Math. Soc. Lect. Note Ser. 98, 1986; Zbl 0589.16014)] shows that either Q is properly contained in P, or else there is a link from Q to P. In the first three sections of the paper, the authors offer a number of sufficient conditions on P, Q, U, and V for the existence of such an extension. As an application, they answer negatively a long outstanding question whether the annihilator of a (Krull) critical right R-module always has to be prime, as is the case when the Noetherian ring R is ideal invariant. In the last two sections, the impact of the link structure of the prime spectrum on the layered structure of an injective indecomposable module is studied, and the layers are described in analogy with the pioneer work for commutative Noetherian rings done by \textit{E. Matlis} [Pac. J. Math. 8, 511-528 (1958; Zbl 0084.266)]. This investigation is naturally restricted to rings satisfying the strong second layer condition, because the link structure is best understood for these rings, a class which is vast enough, as it comprises such rings as group rings of polycyclic-by-finite groups, enveloping algebras of finite-dimensional solvable Lie algebras over algebraically closed fields, Ore and skew polynomial extensions of commutative Noetherian rings, and FBN rings. The best results are obtained for polynormal rings and for group rings of polycyclic-by-finite groups with commutative Noetherian coefficient rings. The additional insight is possible because in these cases the link graphs are locally finite.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 4063294
zbMATH Keywords: Noetherian ring
zbMATH Keywords: non-split extension of a uniform submodule
zbMATH Keywords: prime ideal
zbMATH Keywords: annihilator
zbMATH Keywords: prime spectrum
zbMATH Keywords: injective indecomposable module
zbMATH Keywords: layers
zbMATH Keywords: second layer condition
zbMATH Keywords: group rings of polycyclic-by-finite groups
zbMATH Keywords: enveloping algebras of finite-dimensional solvable Lie algebras
zbMATH Keywords: FBN rings
zbMATH Keywords: link graphs
author: Kenneth A. Brown
author: Robert B. jun. Warfield
reviewed by: Günter Krause
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90219-0
OpenAlex ID: W2083646111
cites work: Localization and ideal theory in Noetherian strongly group-graded rings
cites work: Q3689310
cites work: The structure of modules over polycyclic groups
cites work: Q3037624
cites work: Localisation at cliques in group rings
cites work: On the Representation Theory of Solvable Lie Algebras
cites work: Q3240881
cites work: Quadratic Extensions of Skew Fields
cites work: Non-Artinian Essential Extensions of Simple Modules
cites work: Krull dimension
cites work: Q3717234
cites work: Q3858416
cites work: Q3677877
cites work: Injective modules over Noetherian rings
cites work: Q3876064
cites work: INJECTIVE MODULES FOR GROUP RINGS OF POLYCYCLIC GROUPS I
cites work: Finiteness of the injective hull
cites work: Krull dimensions of the enveloping algebra of sl(2,C)
cites work: Stable structure of noncommutative Noetherian rings. II
cites work: Non-holonomic modules over Weyl algebras and enveloping algebras
cites work: On the Ideals of a Noetherian Ring
Recommended article: Prime ideals in finite extensions of Noetherian rings
Recommended article: A homological reformulation of the link condition.
Recommended article: Cohen’s theorem and eakin-nagata theorem revisited
Recommended article: Q4501426
Recommended article: Q4893002
Recommended article: Q3769053
Recommended article: Noncommutative Noetherian rings. With the cooperation of L. W. Small.
Recommended article: Affiliated series and extensions of modules
Recommended article: On Cohen's theorem for modules
Recommended article: A proof of the theorem of Eakin-Nagata
description: scientific article; zbMATH DE number 4063294