A decomposition theory of comodules (Q1340981)

Let \(C\) be a coalgebra over a field \(k\) and \(M\) be a right \(C\)-comodule. For any \(m \in M\) denote by \(\langle m\rangle\) the minimal closed subcomodule in \(M\) which contains \(m\). It is shown that there exist finitely many maximal ideals \(P\) in \(C^*\) such that \(P^\perp \subseteq \langle m\rangle\). All these ideals are closed. Let \(P_1, \dots, P_r\) be all ideals in \(C^*\) with the property mentioned. There exists a positive integer \(n\) such that \((P_1 \cap \dots \cap P_r)^n \langle m\rangle = 0\). A subcomodule \(N \supseteq M\) is weakly closed if \(x \in N \Rightarrow \langle x \rangle \supseteq N\). A necessary and sufficient condition is found for a comodule \(M\) to be a direct sum of weakly closed indecomposable subcomodules. These decompositions are related to the operation \(X \wedge_C Y = \Delta^{-1} (C \otimes Y + X \otimes C)\) on subcomodules.