Thisexpositorymographwaswrittenforthreereasons. Firstly,wewantedto present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we w 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 phemen in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and t for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does t hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does t hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phemen to a wider ma- ematical audience.
Alberto Facchini is a Professor of Mathematics at the University of Padua.