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# Article

 Title: Relatively exact modules (English) Author: Bican, Ladislav Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 44 Issue: 4 Year: 2003 Pages: 569-576 . Category: math . Summary: Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice $\Cal E_{\tau}(M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application we also obtain some new characterizations of perfect torsion theories. (English) Keyword: Hereditary torsion theory $\tau$ Keyword: $\tau$-injective module Keyword: $\tau$-exact module Keyword: preradical Keyword: exact torsion theory Keyword: perfect torsion theory MSC: 16D50 MSC: 16D80 MSC: 16S90 MSC: 18E40 idZBL: Zbl 1101.16023 idMR: MR2062873 . Date available: 2009-01-08T19:31:13Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119411 . Reference: [1] Anderson F.W., Fuller K.R.: Rings and Categories of Modules.Graduate Texts in Mathematics, vol.13 Springer-Verlag (1974). Zbl 0301.16001, MR 0417223 Reference: [2] Bican L.: Torsionfree precovers.to appear. Zbl 1074.16002, MR 2080845 Reference: [3] Bican L., El Bashir R., Enochs E.: All modules have flat covers.Proc. London Math. Society 33 (2001), 649-652. Zbl 1029.16002, MR 1832549 Reference: [4] Bican L., Torrecillas B.: Precovers.Czechoslovak Math. J. 53 (128) (2003), 191-203. Zbl 1016.16003, MR 1962008 Reference: [5] Bican L., Torrecillas B.: On covers.J. Algebra 236 (2001), 645-650. Zbl 0973.16002, MR 1813494 Reference: [6] Bican L., Torrecillas B.: On the existence of relative exact covers.Acta Math. Hungar. 95 (2002), 178-186. MR 1905180 Reference: [7] Bican L., Torrecillas B.: Relative exact covers.Comment. Math. Univ. Carolinae 42 (2001), 477-487. Zbl 1068.16039, MR 1883369 Reference: [8] Bican L., Kepka T., Němec P.: Rings, Modules, and Preradicals.Marcel Dekker New York (1982). MR 0655412 Reference: [9] Golan J.: Torsion Theories.Pitman Monographs and Surveys in Pure an Applied Mathematics, 29 Longman Scientific and Technical (1986). Zbl 0657.16017, MR 0880019 Reference: [10] Rim S.H., Teply M.L.: On coverings of modules.Tsukuba J. Math. 24 (2000), 15-20. Zbl 0985.16017, MR 1791327 Reference: [11] García Rozas J.R., Torrecillas B.: On the existence of covers by injective modules relative to a torsion theory.Comm. Algebra 24 (1996), 1737-1748. MR 1386494 Reference: [12] Teply M.L.: Torsion-free covers II.Israel J. Math. 23 (1976), 132-136. Zbl 0321.16014, MR 0417245 Reference: [13] Xu J.: Flat Covers of Modules.Lecture Notes in Mathematics 1634 Springer Verlag, Berlin-Heidelberg-New York (1996). Zbl 0860.16002, MR 1438789 .

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