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

By Alberto Facchini

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
• Language: English
• ISBN: 3034803028
• ISBN13: 9783034803021
• Release Date: February 2012
• Publisher: Birkhauser
• Length: 285 Pages
• Weight: 0.93 lbs.
• Dimensions: 0.6" x 6.1" x 9.2"