Reducibility index and sum-reducibility index

Abstract

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