Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules - Couverture rigide

Facchini, Alberto

Couverture rigide

ISBN 10 : 3764359080 ISBN 13 : 9783764359089

Editeur : Birkhauser Verlag AG, 1998

Synopsis

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.

Détails de l'édition

• Éditeur: Birkhauser Verlag AG
• Date d'édition: 1998
• Langue: anglais
• ISBN 10: 3764359080
• ISBN 13: 9783764359089
• Reliure: Relié
• Nombre de pages: 288
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible