Approximations and endomorphism algebras of modules

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This revised and extended edition reflects significant developments in the field since the first edition in 2006. The monograph is divided into two volumes: the first focuses on approximation theory, while the second addresses realization theorems for modules. The complexity of classifying all modules over a general associative ring necessitates limiting classification attempts to specific subcategories. The presence of realization theorems indicates the wild character of these categories, as any reasonable algebra can be isomorphic to the endomorphism algebra of a module from a given subcategory. This leads to pathological direct sum decompositions, complicating classification efforts. To address this, approximation theory has been developed to classify modules by approximating them with those from suitable subcategories. The first volume introduces key classes of modules, including S-complete, pure-injective, Mittag-Leffler, and slender modules, and discusses methods of approximation and recent applications related to tilting and cotilting modules. The second volume presents prediction principles and their applications in proving realization theorems, as well as tools for addressing problems in algebraic topology. The authors emphasize the challenges of classification for modules over general rings and document the wild nature of many module categories through realization theorems. This comprehensive work is suitable for bot