Noetherian rings of injective dimension one which are orders in quasi-Frobenius rings.
type: Item
instance of: scholarly article
title: Noetherian rings of injective dimension one which are orders in quasi-Frobenius rings.
zbMATH Open document ID: 1039.16019
DOI: 10.1016/S0021-8693(03)00461-7
published in: Journal of Algebra
publication date: 2004-01-14T00:00:00Z
review text: The paper deals with Noetherian orders \(R\) in a quasi-Frobenius ring \(Q\). It is shown first that \(\text{id}\, R_R=1\) implies that \(R\) has Krull dimension \(\leq 1\) and that \(R/I\) is quasi-Frobenius for every invertible ideal \(I\) of \(R\). Conversely, it is proved that if \(\text{Kdim}\, R=1\) and every maximal right ideal of \(R\) contains an invertible ideal \(I\) with \(R/I\) quasi-Frobenius, then \(\text{id}\, R_R=1\). If \(R\) is an indecomposable Noetherian ring with \(\text{id}\, R_R=1\) which is integral over its center, the authors show that \(R\) is an order in a quasi-Frobenius ring if and only if the Artinian radical of \(R\) is zero. Some criteria for injective dimension one are also given in case \(R\) is semiprime.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 2026926
Wikidata QID: Q122234253
author: A. W. Chatters
author: C. R. Hajarnavis
MaRDI profile type: Publication
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full work available at URL: s0021-8693(03)00461-7
OpenAlex ID: W1989330339
description: scientific article; zbMATH DE number 2026926