2003/11/28 by Shouchuan Zhang
Mathematics · #math.QA #math.RA #msc:16w30
published as Science in China, 40 (1997), 25-29 · 17pages
arxiv created 2003/11/28 · arxiv updated 2009/12/01
T. Shudo and H. Miyamito \citeSM78 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 \citeXSF94. 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)^*