Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative Faithful Exact Functors and Their Applications to Homological Modules

Xiaolei Zhang, Lei Qiao, Hwankoo Kim

Abstract

The notions of faithfully projective, faithfully flat, and faithfully injective modules--defined as modules for which the three classical homological functors are both faithful and exact--play fundamental roles across various areas of algebra. In this paper, we extend these notions to the setting of $w$-operation theory. By introducing the concept of $w$-faithfully exact functors, we define and investigate the notions of $w$-faithfully projective, $w$-faithfully flat, and $w$-faithfully injective modules. We establish their fundamental properties and demonstrate their effectiveness in generalizing classical results.

Relative Faithful Exact Functors and Their Applications to Homological Modules

Abstract

The notions of faithfully projective, faithfully flat, and faithfully injective modules--defined as modules for which the three classical homological functors are both faithful and exact--play fundamental roles across various areas of algebra. In this paper, we extend these notions to the setting of -operation theory. By introducing the concept of -faithfully exact functors, we define and investigate the notions of -faithfully projective, -faithfully flat, and -faithfully injective modules. We establish their fundamental properties and demonstrate their effectiveness in generalizing classical results.
Paper Structure (7 sections, 39 theorems, 239 equations)

This paper contains 7 sections, 39 theorems, 239 equations.

Table of Contents

  1. introduction
  2. New Perspectives on $w$-Operation Theory
  3. The category $(R,w)\text{-}\mathrm{mod}$ and $w$-faithfully exact functors
  4. $w$-faithfully projective modules
  5. $w$-faithfully flat modules
  6. $w$-faithfully injective modules
  7. Funding

Key Result

Proposition 2.2

Let $M$ be a ${\rm GV}$-torsion-free $R$-module. For $x\in {\rm E}(M)$ the following conditions are equivalent: In particular, via the diagonal embedding one has

Theorems & Definitions (95)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 85 more