Module Theory

Alberto Facchini

ISBN 10: 3034803028 / ISBN 13: 9783034803021

Verlag: Springer Basel, Springer Basel Feb 2012, 2012

Beschreibung

Neuware -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.Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 300 pp. Englisch.

Bestandsnummer des Verkäufers 9783034803021

Bibliografische Details

• Titel: Module Theory
• Verlag: Springer Basel, Springer Basel Feb 2012
• Erscheinungsjahr: 2012
• Sprache: Englisch
• ISBN-10: 3034803028
• ISBN-13: 9783034803021
• Einband: Taschenbuch
• Zustand: Neu

Über diesen Titel

Inhaltsangabe

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.

Über die Autorin bzw. den Autor

Alberto Facchini is a Professor of Mathematics at the University of Padua.