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 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 is both subadditive and pre-additive as an invariant of Mod R . Our main goal is to investigate for modules over pseudo-valuation domains. Specifically, we establish which pseudovaluation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class of pseudo-valuation domains as a union = 1 ∪ 2 ∪ 3 ∪ 4 of suitably defined subclasses, and prove that the property holds if and only if R ∈ 3 ∪ 4 . In that case we can describe the R-modules A where A < . We also show that, for R ∈ 4 , there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.