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Strong Purity and Phantom Morphisms

R. Hafezi, J. Asadollahi, S. Sadeghi, Y. Zhang

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 Slávik.

Strong Purity and Phantom Morphisms

Abstract

Let be a commutative ring and be a multiplicative subset. We introduce and study the concept of -purity based on the notion of -strongly flat modules. The class of -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 -phantom morphisms. We will establish that the -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 Slávik.

Paper Structure

This paper contains 10 sections, 26 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and some facts
  3. Strongly flat modules
  4. Periodic strongly flat modules
  5. Phantom morphisms
  6. $S$-purity
  7. Closedness under extension
  8. $S$-phantom morphisms
  9. Existence of $S$-phantom cover
  10. Optimistic Conjecture

Key Result

Proposition 2.5

Let $R$ be a commutative ring and $S\subset R$ be a multiplicative subset such that $(\mathbb{OC})$ holds for $(R, S)$. Let $M$ be an $R$-module fitting into a pure exact sequence $0\rightarrow M\rightarrow G\rightarrow M\rightarrow 0$ such that $G$ is $S$-strongly flat. Then $M$ is $S$-strongly fla

Theorems & Definitions (67)

  • Definition 2.1
  • Remark 2.3
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Proposition 2.9
  • ...and 57 more