Relative Faithful Exact Functors and Their Applications to Homological Modules

TL;DR

This paper extends the classical concepts of faithfully projective, flat, and injective modules to the context of $w$-operation theory, introducing $w$-faithfully exact functors and modules to generalize homological properties.

Contribution

It introduces the notion of $w$-faithfully exact functors and modules, broadening the scope of classical homological module theory within $w$-operation frameworks.

Findings

Defined $w$-faithfully projective, flat, and injective modules.

Established fundamental properties of $w$-faithfully exact functors.

Generalized classical homological results to $w$-operation setting.

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.