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