Extensions of rational modules.
type: Item
instance of: scholarly article
title: Extensions of rational modules.
zbMATH Open document ID: 1066.16037
DOI: 10.1155/S0161171203203471
author: Niranjan Chakravarthy
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/51437
review text: Summary: For a coalgebra \(C\), the rational functor \(\text{Rat}(-)\colon{\mathcal M}_{C^*}\to{\mathcal M}_{C^*}\) is a left exact preradical whose associated linear topology is the family \({\mathcal F}_C\), consisting of all closed and cofinite right ideals of \(C^*\). It was proved by \textit{D. E. Radford} [J. Algebra 26, 512-535 (1973; Zbl 0272.16012)] that if \(C\) is right \(\mathcal F\)-Noetherian (which means that every \(I\in{\mathcal F}_C\) is finitely generated), then \(\text{Rat}(-)\) is a radical. We show that the converse follows if \(C_1\), the second term of the coradical filtration, is right \(\mathcal F\)-Noetherian. This is a consequence of our main result on \(\mathcal F\)-Noetherian coalgebras which states that the following assertions are equivalent: (i) \(C\) is right \(\mathcal F\)-Noetherian; (ii) \(C_n\) is right \(\mathcal F\)-Noetherian for all \(n\in\mathbb{N}\); and (iii) \({\mathcal F}_C\) is closed under products and \(C_1\) is right \(\mathcal F\)-Noetherian. New examples of right \(\mathcal F\)-Noetherian coalgebras are provided.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 18E40
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16W70
zbMATH DE Number: 2166338
zbMATH Keywords: rational functors
zbMATH Keywords: left exact preradicals
zbMATH Keywords: cofinite right ideals
zbMATH Keywords: coradical filtrations
zbMATH Keywords: Noetherian coalgebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2166338