Approximations and Endomorphism Algebras of Modules

Rüdiger Göbel, Jan Trlifaj (Autoren)

Buch | Hardcover

XXIV, 640 Seiten

2006
De Gruyter (Verlag)
978-3-11-011079-1 (ISBN)

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The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Rüdiger Göbel, University of Duisburg-Essen, Germany; Jan Trlifaj, Charles University, Prague, Czech Republic.

"I strongly recommend the monograph to anyone who is interested in the modern theory of modules."(pruz) in: EMS Newsletter 9/2007 "All in all, I highly recommend the book to everyone interested in cotorsion pairs, approximation theory, realization of algebras or application of set theory to algebra."Gábor Braun in: Zentralblatt MATH 1121/2007 "The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory."L'Enseignement Mathematique 3-4/2006

"I strongly recommend the monograph to anyone who is interested in the modern theory of modules."
(pruz) in: EMS Newsletter 9/2007

"All in all, I highly recommend the book to everyone interested in cotorsion pairs, approximation theory, realization of algebras or application of set theory to algebra."
Gábor Braun in: Zentralblatt MATH 1121/2007

"The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory."
L'Enseignement Mathematique 3-4/2006

Erscheint lt. Verlag	18.7.2006
Reihe/Serie	De Gruyter Expositions in Mathematics ; 41
Verlagsort	Berlin/Boston
Sprache	englisch
Maße	170 x 240 mm
Gewicht	1215 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • Algebra, ideal, module, ring, indecomposable modul • Algebra, ideal, module, ring, indecomposable module • Hardcover, Softcover / Mathematik/Arithmetik, Algebra • HC/Mathematik/Arithmetik, Algebra • Ideal • Indecomposable Module • Modul • Module • Modul (math.) • Ring • Unzerlegbarer Modul
ISBN-10	3-11-011079-2 / 3110110792
ISBN-13	978-3-11-011079-1 / 9783110110791
Zustand	Neuware