Strong Purity and Phantom Morphisms

TL;DR

This paper introduces and studies $S$-purity, $S$-phantom morphisms, and related concepts in module theory over commutative rings, establishing properties of these classes and exploring their connections to the 'Optimistic Conjecture' in the context of phantom maps.

Contribution

It develops the theory of $S$-purity and $S$-phantom morphisms, including their properties and relationships, and investigates an ideal version of the 'Optimistic Conjecture'.

Findings

$S$-pure injective modules form an enveloping class.

The $S$-phantom ideal is precovering and sometimes covering.

Connections between purity, phantom maps, and the 'Optimistic Conjecture' are established.

Abstract

Let $R$ be a commutative ring and $S \subseteq R$ be a multiplicative subset. We introduce and study the concept of $S$-purity based on the notion of $S$-strongly flat modules. The class of $S$-pure injective modules will be studied. We demonstrate that this class is enveloping and explore its closedness under extension. The concept of purity is closely connected to the existence of phantom maps. So we will delve into the study of the $S$-phantom morphisms. We will establish that the $S$-phantom ideal is a precovering ideal and examine the situations where it becomes a covering ideal. Finally, in the last section, we will investigate an ideal version of the `Optimistic Conjecture', raised by Positselski and Sl\'{a}vik.