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    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
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Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules

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

Facchini, Alberto

Paperback

Series: Modern Birkhäuser Classics

Algebra

ISBN10: 3034803028
ISBN13: 9783034803021
Publisher: Springer Nature
Published: Feb 5 2012
Pages: 285
Weight: 0.93
Height: 0.63 Width: 6.14 Depth: 9.21
Language: English
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.

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