Reducibility index and sum-reducibility index

Let [Formula: see text] be a commutative Noetherian ring. For finitely generated text]-module text], Northcott introduced the reducibility index of which is number submodules appearing in an irredundant irreducible decomposition submodule text]. On other hand, for Artinian Macdonald proved that sum-irreducible representation does not depend on choice representation. This called sum-reducibility In former part this paper, we compute where flat homomorphism rings. Especially, localization, polynomial extension, and completion are studied. latter clarify relation among Matlis dual