Stabilization of Kurosh chains
type: Item
instance of: scholarly article
title: Stabilization of Kurosh chains
zbMATH Open document ID: 0568.16006
DOI: 10.1007/BF01161581
author: I. V. L'vov
author: Alexander Sidorov
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let K be a commutative ring with identity, and \({\mathcal A}^ a \)class of K-algebras closed under subobjects and quotient objects. Let \({\mathcal M}\) be a subclass of \({\mathcal A}\), and \(\tilde {\mathcal M}\) the smallest radical subclass of \({\mathcal A}\) containing \({\mathcal M}\). This class can be constructed as the union of the so called Kurosh chain (\({\mathcal M}_{\alpha})\) (indexed by ordinals \(\alpha\geq 0)\) for \({\mathcal M}\). One says that the Kurosh chain (\({\mathcal M}_{\alpha})_{\alpha \geq 0}\) of \({\mathcal M}\) stabilizes at the place \(\alpha\) if \(\alpha\) is the smallest ordinal such that \({\mathcal M}_{\alpha}={\mathcal M}_{\alpha +1}\), and then \(\tilde {\mathcal M}={\mathcal M}_{\alpha}\). The aim of this paper is to prove the following result: If the class \({\mathcal A}\) contains all commutative K- algebras and \(\alpha\) is an ordinal such that \(\alpha\leq \omega\), where \(\omega\) is the first transfinite ordinal, then there exists a subclass \({\mathcal M}\) of \({\mathcal A}\) such that the Kurosh chain of \({\mathcal M}\) stabilizes at the place \(\alpha\). There is a mistake in the definition of the first class \({\mathcal M}_ 0\) in the Kurosh chain (\({\mathcal M}_{\alpha})_{\alpha \geq 0}:\) \({\mathcal M}_ 0\) is the class of all epimorphic images of algebras in \({\mathcal M}\) (and not in \({\mathcal A}\), as is stated in the paper).
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 3906666
zbMATH Keywords: radical subclass
zbMATH Keywords: Kurosh chain
zbMATH Keywords: commutative K-algebras
reviewed by: Toma Albu
MaRDI profile type: Publication
cites work: Q4744417
cites work: A TORELLI THEOREM FOR ALGEBRAIC SURFACES OF TYPEK3
cites work: A Note on Lower Radical Constructions for Associative Rings
cites work: A chain of Kurosh may have an arbitrary finite length
description: scientific article; zbMATH DE number 3906666