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monogeny class; epigeny class; weak Krull-Schmidt theorem; hereditary torsion theory; uniform module; co-uniform module
Recently, A. Facchini [3] showed that the classical Krull-Schmidt theorem fails for serial modules of finite Goldie dimension and he proved a weak version of this theorem within this class. In this remark we shall build this theory axiomatically and then we apply the results obtained to a class of some modules that are torsionfree with respect to a given hereditary torsion theory. As a special case we obtain that the weak Krull-Schmidt theorem holds for the class of modules that are both uniform and co-uniform. A simple example shows that this generalizes the result of [3] mentioned above.
[1] Bican L., Kepka T., Němec P.: Rings, Modules and Preradicals. Marcel Dekker New York, Longman Scientific Publishing, London (1982). MR 0655412
[2] Bican L., Torrecillas B.: QTAG torsionfree modules. Comment. Math. Univ. Carolinae 33 (1994), 1-20. MR 1173740
[3] Facchini A.: Krull-Schmidt fails for serial modules. Trans. Amer. Math. Soc. 348 (1996), 4561-4575. MR 1376546 | Zbl 0868.16003
[4] Golan J.S.: Torsion Theories. Pitman Monographs and Surveys in Pure and Appl. Math. Longman Scientific Publishing, London (1986). MR 0880019 | Zbl 0657.16017
[5] Herbera D., Shamsuddin A.: Modules with semi-local endomorphism rings. Proc. Amer. Math. Soc. 123 (1995), 3593-3600. MR 1277114
[6] Stenström B.: Rings of Quotients. Springer Berlin (1975). MR 0389953
[7] Varadarajan K.: Dual Goldie dimension. Comm. Algebra 7 (1979), 565-610. MR 0524269 | Zbl 0487.16020
[8] Facchini A.: Module Theory. Endomorphism rings and direct decompositions in some classes of modules (Lecture Notes). to appear. MR 1634015
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