Decomposition of comodules (Q1337382)

The purpose of this paper is to give decompositions of comodules. In 1992, the second author [J. Algebra (to appear)] first showed that every restricted comodule can be uniquely decomposed into the direct sum of closed indecomposable subcomodules. In this paper, the authors improve Xu's results and methods and show that for a coalgebra \(C\), every \(C\)- comodule \(M\) can be uniquely decomposed into the direct sum of indecomposable components. A component is in fact a \(C'\)-component of \(M\) for a subcoalgebra \(C'\) of \(C\) which means the sum of all nonzero \(C'\)- subcomodules of \(M\). A \(C\)-comodule is said to be indecomposable iff it cannot be a direct sum of two nonzero components. Since every component is closed, every comodule is uniquely decomposed into the direct sum of closed indecomposable subcomodules. As a special case, a decomposition of every coalgebra follows. A coalgebra is said to be indecomposable iff it cannot be a direct sum of two subcoalgebras. Then every coalgebra can be uniquely decomposed into the direct sum of indecomposable subcoalgebras. Note that the decomposition of \(C\) as \(C\)-comodule is the same as that of \(C\) as coalgebra.