| Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules 1998 Edition Contributor(s): Facchini, Alberto (Author) |
|
 |
ISBN: 3764359080 ISBN-13: 9783764359089 Publisher: Birkhauser OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: June 1998 |
| Additional Information |
| BISAC Categories: - Mathematics | Algebra - General |
| Dewey: 512.4 |
| LCCN: 98022019 |
| Series: Progress in Mathematics |
| Physical Information: 0.75" H x 6.14" W x 9.21" (1.34 lbs) 288 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |