The relation between the decomposition of comodules and coalgebras

Abstract

T. Shudo and H. Miyamito \cite{SM78} showed that $C$ can be decomposed into a direct sum of its indecomposable subcoalgebras of $C$. Y.H. Xu \cite {XF92} showed that the decomposition was unique. He also showed that $M$ can uniquely be decomposed into a direct sum of the weak-closed indecomposable subcomodules of $M$(we call the decomposition the weak-closed indecomposable decomposition) in \cite{XSF94}. In this paper, we give the relation between the two decomposition. We show that if $M$ is a full, $W$-relational hereditary $C$-comodule, then the following conclusions hold: (1) $M$ is indecomposable iff $C$ is indecomposable; (2) $M$ is relative-irreducible iff $C$ is irreducible; (3) $M$ can be decomposed into a direct sum of its weak-closed relative-irreducible subcomodules iff $C$ can be decomposed into a direct sum of its irreducible subcoalgebras. We also obtain the relation between coradical of $C$- comodule $M$ and radical of algebra $C(M)^*$