Precovering and preenveloping ideals

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.Comment: 16 page

Subfunctors of Extension Functors

This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce’s Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. It is showed how Eklof & Trlifaj’s result can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal complement of the ideal. We relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A2 (which is the category of all representations of 2-quiver by R-modules) and obtain a sufficient condition for the existence of covering ideals in a more general setting. We finish with applying this result to the class of phantom morphisms