Decompositions of modules and comodules

Abstract

It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable subrings A = A 1 ⊕ · · · ⊕ A n such that the A i -Mod are indecomposable module categories. Similarly any coalgebra C over a field can be written as a direct sum of indecomposable subcoalgebras C = I C i such that the categories of C i -comodules are indecomposable. In this paper a decomposition theorem for closed subcategories of a module category is proved which implies both results mentioned above as special cases. Moreover it extends the decomposition of coalgebras over fields to coalgebras over noetherian (QF) rings.