Relation between decomposition of comodules and coalgebras
type: Item
instance of: scholarly article
title: Relation between decomposition of comodules and coalgebras
zbMATH Open document ID: 0881.16025
DOI: 10.1007/BF03182866
author: Shouchuan Zhang
published in: Science in China. Series A
publication date: 1998-02-16T00:00:00Z
review text: Let \(M\) be a \(C\)-comodule. It was shown by \textit{T. Shudo} and \textit{H. Miyamoto} [Hiroshima Math. J. 8, 499-504 (1978; Zbl 0412.16005)] that \(C\) can be decomposed into a direct sum of the indecomposable subcoalgebras of \(C\). On the other hand, \textit{Y. Xu}, \textit{K.-P. Shum} and \textit{Y. Fong} [J. Algebra 170, No. 3, 880-906 (1994; Zbl 0827.16029)] showed that the comodule \(M\) can be uniquely decomposed into a direct sum of its weak-closed indecomposable subcomodules. The aim of this paper is to relate these two decompositions. In particular, it is shown that if \(M\) is a \(C\)-comodule, \({\mathcal M}_0=\{N\mid N\) is a minimal closed submodule of \(M\}\) and \({\mathcal C}(M)=\{D\mid D\) is a faithful simple subcoalgebra of \(C\) to \(M\}\), then \(\psi\colon{\mathcal C}(M)\to{\mathcal M}_0\), \(D\mapsto M_D\) is a bijection. Also, it is noted that under certain conditions, the decomposition of \(M\) into the indecomposable comodules of \(M\) is the same as the decomposition of \(C\) into the indecomposable coalgebras of \(C\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1066809
zbMATH Keywords: direct sums
zbMATH Keywords: indecomposable subcoalgebras
zbMATH Keywords: weak-closed indecomposable subcomodules
zbMATH Keywords: indecomposable comodules
zbMATH Keywords: indecomposable coalgebras
reviewed by: K. P. Shum
MaRDI profile type: Publication
cites work: On the decomposition of coalgebras
cites work: Q4347987
cites work: A decomposition theory of comodules
cites work: Q5590089
cites work: Q3933122
cites work: Reflexivity and coalgebras of finite type
full work available at URL: https://doi.org/10.1007/bf03182866
OpenAlex ID: W2015581362
description: scientific article; zbMATH DE number 1066809