Narrow Results

Let $R$ be a commutative 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 $M$.

Let φ: (R, 𝔪) → (S, 𝔫) be a flat local homomorphism of rings. In this paper, we prove: (1) If dim S/𝔪S > 0, then S is a filter ring if and only if R and k(𝔭) ⊗_R𝔭 S_𝔮 are Cohen–Macaulay for all 𝔮 ∈ Spec(S) \ {𝔫} and 𝔭= 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. (2) If dim S/𝔪S = 0, then S is a filter ring if and only if R is a filter ring and k(𝔭) ⊗_R𝔭 S_𝔮 is Cohen–Macaulay for all 𝔮 ∈ Spec(S) \ {𝔫} and 𝔭 = 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. As an application, it is shown that for a k-algebra R and an algebraic field extension K of k, if K ⊗_k R is locally equidimensional, then R is a locally filter ring if and only if K ⊗_k R is a locally filter ring.