Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules (Modern Birkh?user Classics)

Facchini, Alberto

ISBN 10: 3034803028 / ISBN 13: 9783034803021

Published by Birkh?user, 2012

About this Item

This book presents topics in module theory and ring theory: some, such as Goldie dimension and semiperfect rings are now considered classical and others more specialized, such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules. Series: Modern Birkhauser Classics. Num Pages: 298 pages, biography. BIC Classification: PBF. Category: (P) Professional & Vocational. Dimension: 235 x 158 x 17. Weight in Grams: 460. . 2012. 1998th Edition. paperback. . . . .

Seller Inventory # V9783034803021

Bibliographic Details

• Title: Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules (Modern Birkh?user Classics)
• Publisher: Birkh?user
• Publication Date: 2012
• Language: English
• ISBN 10: 3034803028
• ISBN 13: 9783034803021
• Binding: Soft cover
• Condition: New

About this title

Synopsis

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.

About the Author

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