2 8 N ov 2 00 3 The relation between the decomposition of comodules and coalgebras ∗
نویسنده
چکیده
T. Shudo and H. Miyamito [3] showed that C can be decomposed into a direct sum of its indecomposable subcoalgebras of C. Y.H. Xu [5] 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 [6]. 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 Ccomodule M and radical of algebra C(M)∗ 0 Introduction and Preliminaries The decomposition of coalgebras and comodules is an important subject in study of Hopf algebras. T. Shudo and H. Miyamito [3] showed that C can be decomposed into a direct sum of its indecomposable subcoalgebras of C. Y.H. Xu [5] 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 weakclosed indecomposable decomposition ) in [6]. In this paper, we give the relation between ∗This work was supported by the National Natural Science Foundation
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