LEADER 04948nam 2200745Ia 450 001 9910784286503321 005 20230828200949.0 010 $a1-282-19483-6 010 $a9786612194832 010 $a3-11-019972-6 024 7 $a10.1515/9783110199727 035 $a(CKB)1000000000335207 035 $a(EBL)275298 035 $a(OCoLC)476020819 035 $a(SSID)ssj0000104942 035 $a(PQKBManifestationID)11130471 035 $a(PQKBTitleCode)TC0000104942 035 $a(PQKBWorkID)10100371 035 $a(PQKB)10020133 035 $a(MiAaPQ)EBC275298 035 $a(DE-B1597)19800 035 $a(OCoLC)979599439 035 $a(DE-B1597)9783110199727 035 $a(Au-PeEL)EBL275298 035 $a(CaPaEBR)ebr10154725 035 $a(CaONFJC)MIL219483 035 $a(OCoLC)935261822 035 $a(EXLCZ)991000000000335207 100 $a20060602d2006 uy 0 101 0 $aeng 135 $aurnn#---|u||u 181 $ctxt 182 $cc 183 $acr 200 10$aApproximations and endomorphism algebras of modules$b[electronic resource] /$fby Ru?diger Go?bel and Jan Trlifaj 210 $aBerlin ;$aNew York $cWalter de Gruyter$d2006 215 $a1 online resource (664 p.) 225 1 $aDe Gruyter expositions in mathematics ;$v41 300 $aDescription based upon print version of record. 311 0 $a3-11-011079-2 320 $aIncludes bibliographical references and index. 327 $tFront matter --$tContents --$tChapter 1. Some useful classes of modules --$tChapter 2. Approximations of modules --$tChapter 3. Complete cotorsion pairs --$tChapter 4. Deconstruction of cotorsion pairs --$tChapter 5. Tilting approximations --$tChapter 6. 1-tilting modules and their applications --$tChapter 7. Tilting approximations and the finitistic dimension conjectures --$tChapter 8. Cotilting modules --$tChapter 9. The Black Box and its relatives --$tChapter 10. Independence results for cotorsion pairs --$tChapter 11. The lattice of cotorsion pairs --$tChapter 12. Realizing algebras - by algebraically independent elements and by prediction principles --$tChapter 13. E(R)-algebras --$tChapter 14. Modules with distinguished submodules --$tChapter 15. Some useful classes of algebras --$tBackmatter 330 $aThe category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory. 410 0$aGruyter expositions in mathematics ;$v41. 606 $aModules (Algebra) 606 $aModuli theory 606 $aApproximation theory 610 $aAlgebra, ideal, module, ring, indecomposable module. 615 0$aModules (Algebra) 615 0$aModuli theory. 615 0$aApproximation theory. 676 $a512/.42 686 $a510$2sdnb 686 $aSK 150$2rvk 686 $aSK 230$2rvk 686 $aSK 820$2rvk 700 $aGo?bel$b R$g(Ru?diger),$f1940-$060070 701 $aTrlifaj$b Jan$01561252 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910784286503321 996 $aApproximations and endomorphism algebras of modules$93827819 997 $aUNINA