Precovering and preenveloping ideals

TL;DR

This paper explores the extension of cotorsion pair concepts to ideal approximation theory in abelian categories, focusing on conditions when ideals are generated by sets of homomorphisms, inspired by Eklof-Trlifaj results.

Contribution

It generalizes cotorsion pair theory to ideals in abelian categories, establishing conditions for when ideals generated by sets of homomorphisms are precovering and preenveloping.

Findings

Identifies conditions for ideals to be precovering and preenveloping

Extends cotorsion pair results to ideal approximation theory

Provides new insights into ideal generation by sets of homomorphisms

Abstract

L. Salce introduced the notion of a cotorsion pair (F,C) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (F,C) of classes of R-modules is cogenerated by a set, then it is complete. Recently Herzog, Fu, Asensio and Torrecillas developed the ideal approximation theory. In this article we look at a result motivated by the Eklof-Trlifaj argument for an ideal I when it is generated by a set of homomorphisms.