| Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules 1998 Edition Contributor(s): Facchini, Alberto (Author) |
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ISBN: 3034803028 ISBN-13: 9783034803021 Publisher: Birkhauser OUR PRICE: $80.74 Product Type: Paperback - Other Formats Published: February 2012 |
| Additional Information |
| BISAC Categories: - Mathematics | Algebra - General |
| Dewey: 512.4 |
| Series: Modern Birkhäuser Classics |
| Physical Information: 0.63" H x 6.14" W x 9.21" (0.93 lbs) 285 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |