Über N. I. Dubrovin's Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen. (On N. I. Dubrovin's attempt to the construction of not completely prime prime ideals in chain rings)
type: Item
instance of: scholarly article
title: Über N. I. Dubrovin's Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen. (On N. I. Dubrovin's attempt to the construction of not completely prime prime ideals in chain rings)
zbMATH Open document ID: 0713.16003
DOI: 10.1007/BF03322466
author: Martin Schröder
published in: Results in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: A chain ring R is a ring with identity for which aR\(\subseteq bR\) or bR\(\subseteq aR\) holds for any a,b in R and the corresponding condition for left ideals. These rings appear in geometry, module theory and as noncommutative valuation rings in division algebras. The present article begins with a review of papers of the past twenty-five years in which the following question was encountered: Does there exist a chain ring R with a prime ideal P that is not completely prime; i.e. AB\(\subseteq P\), \(A\varsubsetneq P\) implies \(B\subseteq P\) for ideals A,B of R, but there exist elements a,b in R with ab\(\in P\) and both a,b\(\not\in P\). The question remains open and it is the purpose of this paper to point out that the gap in the construction of an example for such a ring as suggested by \textit{N. I. Dubrovin} [Math. USSR, Sb. 48, 437-444 (1984), in particular: Lemma 3, p. 439; translation from Mat. Sb., Nov. Ser. 120, 441-447 (1983; Zbl 0517.16002)] cannot be easily filled. The ring in that example was constructed as a localization (in the sense of P. M. Cohn) of some subring of a certain group ring. Here it is shown by a rather ingenious example that a crucial property is not preserved under localization as was assumed in the above mentioned paper.
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16U20
zbMATH DE Number: 4173161
zbMATH Keywords: chain ring
zbMATH Keywords: left ideals
zbMATH Keywords: noncommutative valuation rings in division algebras
zbMATH Keywords: prime ideal
zbMATH Keywords: completely prime
zbMATH Keywords: localization
reviewed by: H. H. Brungs
MaRDI profile type: Publication
cites work: Group rings and generalized valuations
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full work available at URL: https://doi.org/10.1007/bf03322466
OpenAlex ID: W2075243095
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description: scientific article; zbMATH DE number 4173161