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Module Theory Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules
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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. The problem remained open for 63 years: its solution, a negative 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 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 Warfield'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 math- ematical audience.
Publisher Name | Birkhauser |
---|---|
Author Name | Hagendorf, Col |
Format | Audio |
Bisac Subject Major | MAT |
Language | NG |
Isbn 10 | 3764359080 |
Isbn 13 | 9783764359089 |
Target Age Group | min:NA, max:NA |
Series | 000019240167 |
Dimensions | 00.92" H x 10.06" L x 14.00" W |
Page Count | 288 |
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