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

ISBN-13: 9783034803021 / Angielski / Miękka / 2012 / 285 str.

Alberto Facchini

Opis

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.

Spis treści

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.

O autorze

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

Blurb

The purpose of this expository monograph is three-fold. First, the solution of a problem posed by Wolfgang Krull in 1932 is presented. He asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. A negative answer was published only in 1995 by Facchini, Herbera, Levy and Vámos. Second, the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt-Theorem holds for serial modules, is described. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomena so rare in the history of Krull-Schmidt type theorems that its presentation to a wider mathematical audience provides the third incentive for this monograph. Briefly, the Krull-Schmidt-Theorem holds for some, not all, classes of modules. When it does hold, any two indecomposable decompositions are uniquely determined up to one permutation. For serial modules the theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations.

Apart from these issues, the book addresses various topics in module theory and ring theory, some now considered classical (such as Goldie dimension, semiperfect rings, Krull dimension, rings of quotients, and their applications) and others more specialized (such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules, exchange property, (sigma)-pure-injective modules). Open problems conclude the work.

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Besides its research value, this book is a considerable addition to the list of fundamental books in rings and modules. It is written carefully with necessary backgrounds developed in a logical way. There are many fundamental points worked out in the book. All these make the book an excellent contribution to the development of module and ring theory, and a source of reference. Algebraists will certainly enjoy themselves in reading this book.

(Zentralblatt MATH)

This excellent book on module and ring theory (…) contains three kind of topics: classical topics, (…) specialized topics, (…) the solutions to two famous problems, together with the presentation of a rare phenomenon.

(Mathematica)