Flat-precover completing domains

TL;DR

This paper introduces the concept of flat-precover completing domains, a new framework for analyzing module flatness relative to precovering classes, generalizing classical results and characterizations in ring theory.

Contribution

It defines flat-precover completing domains and extends classical module and ring properties within this new framework.

Findings

Generalizes known results on module flatness.

Provides characterizations of classical rings using these domains.

Introduces a unified approach to module properties.

Abstract

Recently, many authors have embraced the study of certain properties of modules such as projectivity, injectivity and flatness from an alternative point of view. Rather than saying a module has a certain property or not, each module is assigned a relative domain which, somehow, measures to which extent it has this particular property. In this work, we introduce a new and fresh perspective on flatness of modules. However, we will first investigate a more general context by introducing domains relative to a precovering class $\x$. We call these domains $\x$-precover completing domains. In particular, when $\x$ is the class of flat modules, we call them flat-precover completing domains. This approach allows us to provide a common frame for a number of classical notions. Moreover, some known results are generalized and some classical rings are characterized in terms of these domains.