# Article

 Title: Precovers and Goldie’s torsion theory (English) Author: Bican, Ladislav Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 128 Issue: 4 Year: 2003 Pages: 395-400 Summary lang: English . Category: math . Summary: Recently, Rim and Teply , using the notion of $\tau$-exact modules, found a necessary condition for the existence of $\tau$-torsionfree covers with respect to a given hereditary torsion theory $\tau$ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau$-torsionfree and $\tau$-exact covers have been investigated in . The purpose of this note is to show that if $\sigma = (\mathcal T_{\sigma },\mathcal F_{\sigma })$ is Goldie’s torsion theory and $\mathcal F_{\sigma }$ is a precover class, then $\mathcal F_{\tau }$ is a precover class whenever $\tau \ge \sigma$. Further, it is shown that $\mathcal F_{\sigma }$ is a cover class if and only if $\sigma$ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that $\mathcal F_{\tau }$ is a cover class for all hereditary torsion theories $\tau \ge \sigma$. (English) Keyword: hereditary torsion theory Keyword: Goldie’s torsion theory Keyword: non-singular ring Keyword: precover class Keyword: cover class Keyword: torsionfree covers Keyword: lattices of torsion theories MSC: 16D80 MSC: 16D90 MSC: 16S90 MSC: 18E40 idZBL: Zbl 1057.16027 idMR: MR2032476 DOI: 10.21136/MB.2003.134006 . Date available: 2009-09-24T22:11:09Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134006 . Reference: [1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules.Graduate Texts in Mathematics, vol. 13, Springer, 1974. MR 0417223 Reference: [2] L. Bican, B. Torrecillas: Precovers.Czechoslovak Math. J. 53 (2003), 191–203. MR 1962008 Reference: [3] L. Bican, B. Torrecillas: On covers.J. Algebra 236 (2001), 645–650. MR 1813494, 10.1006/jabr.2000.8562 Reference: [4] L. Bican, R. El Bashir, E. Enochs: All modules have flat covers.Proc. London Math. Society 33 (2001), 385–390. MR 1832549 Reference: [5] L. Bican, B. Torrecillas: Relative exact covers.Comment. Math. Univ. Carolinae 42 (2001), 601–607. MR 1883369 Reference: [6] L. Bican, T. Kepka, P. Němec: Rings, Modules, and Preradicals.Marcel Dekker, New York, 1982. MR 0655412 Reference: [7] J. Golan: Torsion Theories.Pitman Monographs and Surveys in Pure an Applied Matematics, 29, Longman Scientific and Technical, 1986. Zbl 0657.16017, MR 0880019 Reference: [8] S. H. Rim, M. L. Teply: On coverings of modules.Tsukuba J. Math. 24 (2000), 15–20. MR 1791327, 10.21099/tkbjm/1496164042 Reference: [9] M. L. Teply: Torsion-free covers II.Israel J. Math. 23 (1976), 132–136. Zbl 0321.16014, MR 0417245 Reference: [10] J. Xu: Flat Covers of Modules.Lecture Notes in Mathematics 1634, Springer, Berlin, 1996. Zbl 0860.16002, MR 1438789 .

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