Module Theory : Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules Paperback
Part of the Modern Birkhauser Classics series
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.
- Format: Paperback
- Pages: 285 pages, XIII, 285 p.
- Publisher: Springer Basel
- Publication Date: 04/02/2012
- Category: Algebra
- ISBN: 9783034803021
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