\(\Omega\)-Krull rings and Gr-\(\Omega\)-Krull rings in the non-P.I. case
type: Item
instance of: scholarly article
title: \(\Omega\)-Krull rings and Gr-\(\Omega\)-Krull rings in the non-P.I. case
zbMATH Open document ID: 0541.16006
DOI: 10.1016/0022-4049(84)90017-3
author: P. Wauters
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: The class of \(\Omega\)-Krull rings consists of noncommutative rings possessing properties generalizing those of Krull domains. The prefix \(\Omega\) refers to \(\Omega\)-rings which had been mentioned earlier by E. Nauwelaerts, F. Van Oystaeyen and \(\Omega\) refers to the fact that these rings are characterized by conditions on the maximal ideals. One may also give an intrinsic graded definition when the rings considered are graded, mostly by a torsion free abelian group or cancellative monoid here, and doing this one obtains the class of Gr-\(\Omega\)-Krull rings. The main result of this note provides sufficient and necessary conditions for a graded ring R to be a central \(\Omega\)-Krull ring in terms of the graded property for R and the ungraded property of the graded symmetric ring of quotients of R. The author applies this result to the study of certain special generalized Rees rings and in particular he derives conditions for a (twisted) group ring over a torsion free abelian group to be an \(\Omega\)-Krull ring. In the P.I. case \(\Omega\)-Krull rings will be just Krull orders and then other and more specific results are known. The use of the symmetric ring of quotients allows to obtain results in the non-PI case and this includes some examples over simple Noetherian non-Artinian rings.
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3859328
zbMATH Keywords: \(\Omega\)-Krull rings
zbMATH Keywords: Krull domains
zbMATH Keywords: maximal ideals
zbMATH Keywords: graded ring
zbMATH Keywords: central \(\Omega\)-Krull ring
zbMATH Keywords: graded symmetric ring of quotients
zbMATH Keywords: generalized Rees rings
zbMATH Keywords: Krull orders
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90017-3
OpenAlex ID: W2092087958
cites work: Graded krull domains
cites work: Krull Semigroups and Divisor Class Groups
cites work: A characterization of central \(\Omega\)-Krull rings
cites work: Generalized Rees rings and relative maximal orders satisfying polynomial identities
cites work: Ordres maximaux au sens de K. Asano
cites work: Lectures on unique factorization domains. Notes by M. Pavman Murthy
cites work: Integral extensions of rings satisfying a polynomial identity
cites work: Q3944732
cites work: Generalized Rees rings and arithmetical graded rings
cites work: Q3663461
description: scientific article; zbMATH DE number 3859328