On Tight Submodules of Modules over Valuation Domains

TL;DR

This paper investigates a specific subclass of modules over valuation domains with projective dimension at most one, focusing on their properties like injectivity and cotorsionness, using a simplified approach based on existing module theory.

Contribution

It introduces and analyzes a novel subclass of modules over valuation domains, highlighting their basic features and properties without a full categorical framework.

Findings

Modules in this subclass share properties with modules over rank one discrete valuation domains.

The study simplifies the understanding of injectivity, pure-injectivity, and cotorsionness in this context.

Several features are easier to analyze due to the restricted nature of the subclass.

Abstract

This note offers an unusual approach of studying a class of modules inasmuch as it is investigating a subclass of the category of modules over a valuation domain. This class is far from being a full subcategory, it is not even a category. Our concern is the subclass consisting of modules of projective dimension not exceeding one, admitting only morphisms whose kernels and cokernels are also objects in this subclass. This class is still tractable, several features are in general simpler than in module categories, but lots of familiar properties are lost. A number of results on modules in this class are similar to those on modules over rank one discrete valuation domains (where the global dimension is 1). The study is considerably simplified by taking advantage of the general theory of modules over valuation domains available in the literature, e.g. in [14]-[15]. Our main goal is to establish the basic features and have a closer look at injectivity, pure-injectivity, and cotorsionness, but we do not wish to enter into an in-depth study of these properties.