Approximations and endomorphism algebras of modules

The category of all modules over a general associative ring is too complex for reasonable classification, especially if the ring is not of finite representation type. Classification efforts must therefore focus on restricted subcategories of modules. The wild nature of these categories is often highlighted by realization theorems, which indicate that any reasonable algebra can be isomorphic to the endomorphism algebra of a module from a specific subcategory. This leads to problematic direct sum decompositions, complicating classification efforts. Realization theorems serve as key indicators of the non-classification theory of modules. To address these challenges, approximation theory has emerged, allowing for the selection of suitable subcategories whose modules can be classified, while approximating arbitrary modules by those from these subcategories. Although these approximations are generally neither unique nor functorial, they provide a wealth of options tailored to various applications. This monograph integrates approximation theory with realization theorems, beginning with foundational concepts and progressing to advanced topics, including recent applications to infinite dimensional tilting theory. It is designed for both graduate students in algebra and experts in module and representation theory.