Ideal arithmetic in Noetherian PI rings
type: Item
instance of: scholarly article
title: Ideal arithmetic in Noetherian PI rings
zbMATH Open document ID: 0671.16004
DOI: 10.1016/0021-8693(89)90230-5
author: A. W. Chatters
author: C. R. Hajarnavis
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: If R is a commutative noetherian domain then every ideal of R is a product of prime ideals if and only if every non-zero ideal of R is invertible. In this paper, the authors study noncommutative versions of this result. A (noncommutative) ring R is a Dedekind prime ring if it is noetherian, hereditary and every non-zero ideal is invertible. The authors prove that a prime noetherian PI ring R is a Dedekind prime ring if and only if every ideal is a product of maximal ideals and that R is then a finite module over its centre which is a Dedekind domain. A key point is that a one-sided invertible ideal is both right and left localisable - this depends on a recent symmetry result of \textit{A. Braun} and \textit{R. B. Warfield} jun. [J. Algebra 118, 322-335 (1988; Zbl 0658.16008)].
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 4097569
zbMATH Keywords: product of prime ideals
zbMATH Keywords: Dedekind prime ring
zbMATH Keywords: prime noetherian PI ring
zbMATH Keywords: product of maximal ideals
zbMATH Keywords: Dedekind domain
zbMATH Keywords: invertible ideal
reviewed by: T. H. Lenagan
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90230-5
OpenAlex ID: W2070156765
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description: scientific article; zbMATH DE number 4097569