Relation between decomposition of comodules and coalgebras
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 \(C\).
scientific article; zbMATH DE number 1066809