t-Dual Baer Modules and t-Lifting Modules

TL;DR

This paper introduces the concepts of t-dual Baer and t-lifting modules, establishing their properties and equivalences over certain rings, thereby generalizing classical module theory notions.

Contribution

It defines t-dual Baer and t-lifting modules and proves their equivalence under specific conditions, extending the theory of lifting modules.

Findings

An amply supplemented module is t-lifting iff it is t-dual Baer and a t-\mathcal{K}-module.

Over a right perfect ring, all modules being t-dual Baer, t-lifting, and injective modules being t-lifting are equivalent.

The paper generalizes classical module concepts through the introduction of t-structures.

Abstract

We introduce the notions of t-lifting modules and t-dual Baer modules, which are generalizations of lifting modules. It is shown that an amply supplemented module $M$ is t-lifting if and only if $M$ is t-dual Baer and a t-$\mathcal{K}$-module. We also prove that, over a right perfect ring $R$, every noncosingular $R$-module is injective if and only if every $R$-module is t-dual Baer if and only if every $R$-module is t-lifting if and only if every injective $R$-module is t-lifting.