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

Alberto Facchini

ISBN 10: 3764359080 ISBN 13: 9783764359089

Editorial: Birkhäuser, 1998

Idioma: Ingl�s

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Detalles bibliogr�ficos

T�tulo

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

Editor

Birkh�user

Fecha de publicaci�n

1998

Idioma

Ingl�s

ISBN 10

3764359080

ISBN 13

9783764359089

Encuadernaci�n

Buch

Estado

Neu

Dimensiones

Peso

629 gramos

Cat�logos de vendedores

B�cher

Acerca de

Sinopsis

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.

Cr�ticas

"Written in an attractive and fresh mathematical style. Each topic is arranged well and lucidly. The author has made important contributions to the study of direct sum decompositions and many of the main results in this book include his own work."

--EMS Newsletter

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