Let $R$ $R$ be a commutative Noetherian ring. For a finitely generated $R$ $R$ -module $M$ $M$ , Northcott introduced the reducibility index of $M$ $M$ , which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ $0$ in $M$ $M$ . On the other hand, for an Artinian $R$ $R$ -module $A$ $A$ , Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$ $A$ . In the former part of this paper, we compute the reducibility index of $S \otimes_{R} M$ $S \otimes_{R} M$ , where $R \to S$ $R \to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$ $M$ , that of the completion of $M$ $M$ , and the sum-reducibility index of the Matlis dual of $M$ $M$ .

Communicated by B. Olberding

Keywords:

AMSC: 13A15, 13H10

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