Home » Module Theory: Endomorphism Rings And Direct Sum Decompositions In Some Classes Of Modules by Alberto Facchini
Module Theory: Endomorphism Rings And Direct Sum Decompositions In Some Classes Of Modules Alberto Facchini

Module Theory: Endomorphism Rings And Direct Sum Decompositions In Some Classes Of Modules

Alberto Facchini

Published June 16th 1998
ISBN : 9783764359089
Hardcover
288 pages
Enter the sum

 About the Book 

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the Krull-Schmidt Theorem holds for ar tinian modules.MoreThis expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the Krull-Schmidt Theorem holds for ar tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krulls question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield 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 so lution to Warfields 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 math ematical audience.