Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules by Alberto FacchiniModule Theory: Endomorphism rings and direct sum decompositions in some classes of modules by Alberto Facchini

Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules

byAlberto Facchini

Paperback | February 5, 2012

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Thisexpositorymonographwaswrittenforthreereasons. Firstly,wewantedto present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-SchmidtTheorem" holds for - tinianmodules. Theproblemremainedopenfor63years:itssolution,anegative answer to Krull's question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). ´ Secondly, we wanted to present the answer to a question posed by War?eld in 1975 [War?eld 75]. He proved that every ?nitely p- sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, War?eld asked whether the "Krull-Schmidt Theorem" holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the - lution to War?eld's problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider ma- ematical audience.
Alberto Facchini is a Professor of Mathematics at the University of Padua.
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Title:Module Theory: Endomorphism rings and direct sum decompositions in some classes of modulesFormat:PaperbackDimensions:285 pages, 23.5 × 15.5 × 0.01 inPublished:February 5, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034803028

ISBN - 13:9783034803021

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Table of Contents

Basic Concepts.- The Krull-Schmidt-Remak-Azumaya Theorem.- Semiperfect Rings.- Semilocal Rings.- Serial Rings.- Quotient Rings.- Krull Dimension and Serial Rings.- Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules.- Biuniform Modules.- ?-pure-injective Modules and Artinian Modules.- Open Problems.