Localization and ideal theory in Noetherian strongly group-graded rings
type: Item
instance of: scholarly article
title: Localization and ideal theory in Noetherian strongly group-graded rings
zbMATH Open document ID: 0607.16013
DOI: 10.1016/0021-8693(87)90181-5
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: A right Noetherian ring R satisfies the right second layer condition if, for every prime ideal P of R, whenever M is a finitely generated uniform right R-module which contains a copy of a non-zero right ideal of R/P and has prime annihilator Q, we have \(P=Q\). This condition is crucial to the theory of localization at (possibly infinite) sets of prime ideals of R and to the representation theory of R. It is known to hold in many (though not all) Noetherian rings. Details can be found in [\textit{A. V. Jategaonkar}, Localization in Noetherian rings (Lond. Math. Soc. Lect. Note Ser. 98, 1986; Zbl 0589.16014)]. The main result of this paper (7.4), states that if G is a polycyclic-by- finite group and S is a strongly G-graded ring with right Noetherian base ring R, then S has the right second layer condition provided R is right Artinian, or G-simple, or satisfies a polynomial identity. This partially generalizes earlier work of Jategaonkar and the reviewer on ordinary group rings - consult the above reference. In developing the background results needed to prove this result, the author provides an overview of much of what is known about rings strongly graded by a polycyclic-by- finite group, and obtains several interesting subsidiary results.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3981376
zbMATH Keywords: right second layer condition
zbMATH Keywords: finitely generated uniform right R-module
zbMATH Keywords: prime annihilator
zbMATH Keywords: localization
zbMATH Keywords: prime ideals
zbMATH Keywords: Noetherian rings
zbMATH Keywords: polycyclic-by-finite group
zbMATH Keywords: strongly G-graded ring
author: Allen D. Bell
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90181-5
OpenAlex ID: W2005772328
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