01690nam a2200217Ia 4500 CTU_183634 210402s9999 xx 000 0 und d 1455000 512.4 F137 Facchini, Alberto Module theory : Endomorphism rings and direct sum decompositions in some classes of modules Alberto Facchini Boston Birkhauser 2012 The purpose of this expository monograph is three-fold. First, the solution of a problem posed by Wolfgang Krull in 1932 is presented. He asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. A negative answer was published only in 1995 by Facchini, Herbera, Levy and Vámos. Second, the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt-Theorem holds for serial modules, is described. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomenon so rare in the history of Krull-Schmidt type theorems that its presentation to a wider mathematical audience provides the third incentive for this monograph. Briefly, the Krull-Schmidt-Theorem holds for some, not all, classes of modules. When it does hold, any two indecomposable decompositions are uniquely determined up to one permutation. For serial modules the theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. Modules (Algebra),Modules (Đại số học) Hải Trung tâm Học liệu Trường Đại học Cần Thơ