The trace property in preenveloping classes

TL;DR

This paper develops the theory of trace modules within preenveloping classes, identifying new examples and characterizing rings with Gorenstein and regular properties, advancing understanding of module trace properties.

Contribution

It introduces a comprehensive framework for trace modules in preenveloping classes and explores their relationship, providing new examples and characterizations of rings.

Findings

New examples of trace ideals and modules identified

Characterizations of Gorenstein and regular rings based on trace properties

Enhanced understanding of the relationship between preenveloping classes and trace modules

Abstract

We develop the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we identify new examples of trace ideals and trace modules, and characterize several classes of rings with a focus on the Gorenstein and regular properties.