Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules (Progress in Mathematics, 167, Band 167) - Hardcover

Facchini, Alberto

Hardcover

ISBN 10: 3764359080 ISBN 13: 9783764359089

Verlag: Birkhäuser, 1998

Inhaltsangabe

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-Schmidt Theorem" holds for ar­ tinian modules. The problem remained open for 63 years: its solution, a negative 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 Warfield in 1975 [Warfield 75]. He proved that every finitely pre­ sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield 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 so­ lution to Warfield'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 math­ ematical audience.

Über diesen Titel

Über die Autorin bzw. den Autor

Alberto Facchini studied at the University of Padua, getting the Laurea in Matematica magna cum laude in 1977. He has worked at the University of Udine from 1981 to 1999, becoming full professor of Algebra, and then at the University of Padua from 1999 to 2022. At present he is Emeritus Professor at the University of Padua. He also got fellowships to study and do research at the University of Sheffield, Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica. He has written about 180 scientific publications and has given lectures and talks at conferences in about forty different countries. In 1981 he received the Bonavera Prize in Mathematics. He has been in the editorial boards of about fifteen mathematical journal, among which Bollettino dell'Unione Matematica Italiana, Communications in Algebra, Journal of Algebra and Its Applications, Rendiconti del Circolo Matematico di Palermo, and Rendiconti del Seminario Matematico dell'Università di Padova.

Details zur Ausgabe

• Verlag: Birkhäuser
• Erscheinungsdatum: 1998
• Sprache: Englisch
• ISBN 10: 3764359080
• ISBN 13: 9783764359089
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 301
• Kontakt zum Hersteller: preigu GmbH & Co. KG; mail@preigu.de; ; Lengericher Landstr. 19; Osnabrück; Niedersachsen; 49078; Deutschland