{"product_id":"module-theory-von-alberto-facchini","title":"Module Theory","description":"\u003cp\u003eThisexpositorymonographwaswrittenforthreereasons. 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.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783034803021\"\u003e\u003ch3\u003eEndomorphism rings and direct sum decompositions in some classes of modules\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783034803021","offer_id":40570955497565,"sku":"9783034803021","price":80.24,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/8eed1b22-093b-4439-8820-66cdc2f4d534.jpg?v=1772170709","url":"https:\/\/shop.autorenwelt.de\/products\/module-theory-von-alberto-facchini","provider":"Autorenwelt Shop","version":"1.0","type":"link"}