The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Abstract

Let R be a commutative ring and A be an R-module. The Mal'cev rank μ(A) of A is the sup of genN, where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N. We prove that μ is both sub-additive and pre-additive as an invariant of Mod(R). Our main goal is to investigate μ for modules over pseudo-valuation domains. Specifically, we establish which pseudo-valuation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class 𝒞 of pseudo-valuation domains as a union 𝒞 = 𝒞1 ∪ 𝒞2 ∪ 𝒞3 ∪ 𝒞4 of suitably defined subclasses, and prove that the property holds if and only if R ∈ 𝒞3 ∪ 𝒞4. In that case we can describe the R-modules A where μ(A) < ∞. We also show that, for R ∈ 𝒞4, there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.