# On subinjectivity domains of pure-injective modules

## TL;DR

This paper explores the concept of pi-indigent pure-injective modules, analyzing their subinjectivity domains and the ring structures where modules exhibit minimal subinjectivity, advancing the understanding of module injectivity properties.

## Contribution

It introduces and studies pi-indigent pure-injective modules and characterizes rings where modules have minimal subinjectivity domains, extending recent module theory research.

## Key findings

- Characterization of subinjectivity domains of pure-injective modules
- Identification of rings where modules are injective or subinjective relative to minimal classes
- Analysis of the structure of rings with minimal subinjectivity conditions

## Abstract

As an alternative perspective on the injectivity of a pure-injective module, a pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module M is called subinjective relative to a module N if for every extension K of N, every homomorphism N \to M can be extended to a homomorphism K \to M. The subinjectivity domain of the module M is defined to be the class of modules N such that M is N-subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigent modules are studied. The structure of a ring over which every pure-injective (resp. simple, uniform, indecomposable) module is injective or subinjective relative only to the smallest possible family of modules is investigated. This work is a natural continuation to recent papers that have embraced the systematic study of the subinjective and subprojective domains of modules.