{"product_id":"9783764359089","title":"Progress in Mathematics: Endomorphism rings and direct sum decompositions in some classes of modules","description":"\u003ch1\u003eProgress in Mathematics: Endomorphism rings and direct sum decompositions in some classes of modules\u003c\/h1\u003e \u003ch2\u003eFacchini, Alberto\u003c\/h2\u003e \u003cp\u003eThis 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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Birkhäuser\u003c\/p\u003e \u003cp\u003ePublication Date: 1998-06-16\u003c\/p\u003e \u003cp\u003eFormat: Hardcover\u003c\/p\u003e \u003cp\u003eISBN-13: 9783764359089\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-3-0348-8774-8\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 288\u003c\/p\u003e ","brand":"Birkhäuser Basel","offers":[{"title":"Default Title","offer_id":46541443170444,"sku":"9783764359089","price":98.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9783764359089.jpg?v=1775736339","url":"https:\/\/lateknightbooks.com\/products\/9783764359089","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}