# Article

 Title: Non-singular covers over monoid rings (English) Author: Bican, Ladislav Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 133 Issue: 1 Year: 2008 Pages: 9-17 Summary lang: English . Category: math . Summary: We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever we replace the strongly cancellative monoid $G$ by the totally ordered cancellative monoid or by the totally ordered group. (English) Keyword: hereditary torsion theory Keyword: torsion theory of finite type Keyword: Goldie’s torsion theory Keyword: non-singular module Keyword: non-singular ring Keyword: monoid ring Keyword: precover class Keyword: cover class MSC: 06F05 MSC: 16D50 MSC: 16D80 MSC: 16S36 MSC: 16S90 MSC: 18E40 idZBL: Zbl 1170.16022 idMR: MR2400147 DOI: 10.21136/MB.2008.133940 . Date available: 2009-09-24T22:33:51Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/133940 . Reference: [1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules.Graduate Texts in Mathematics 13, Springer, 1974. MR 1245487 Reference: [2] L. Bican: Torsionfree precovers.Contributions to General Algebra 15, Proceedings of the 66th Workshop on General Algebra, Klagenfurt 2003, Verlag Johannes Heyn, Klagenfurt, 2004, pp. 1–6. Zbl 1074.16002, MR 2080845 Reference: [3] L. Bican: Precovers and Goldie’s torsion theory.Math. Bohem. 128 (2003), 395–400. Zbl 1057.16027, MR 2032476 Reference: [4] L. Bican: On torsionfree classes which are not precover classes.(to appear). Zbl 1166.16013, MR 2411109 Reference: [5] L. Bican: Non-singular precovers over polynomial rings.Comment. Math. Univ. Carol. 47 (2006), 369–377. Zbl 1106.16032, MR 2281000 Reference: [6] L. Bican: Non-singular covers over ordered monoid rings.Math. Bohem. 131 (2006), 95–104. Zbl 1111.16029, MR 2211006 Reference: [7] L. Bican, R. El Bashir, E. Enochs: All modules have flat covers.Proc. London Math. Society 33 (2001), 385–390. MR 1832549 Reference: [8] L. Bican, B. Torrecillas: Precovers.Czech. Math. J. 53 (2003), 191–203. MR 1962008 Reference: [9] L. Bican, B. Torrecillas: On covers.J. Algebra 236 (2001), 645–650. MR 1813494, 10.1006/jabr.2000.8562 Reference: [10] L. Bican, T. Kepka, P. Němec: Rings, Modules, and Preradicals.Marcel Dekker, New York, 1982. MR 0655412 Reference: [11] J. Golan: Torsion Theories.Pitman Monographs and Surveys in Pure an Applied Mathematics 29, Longman Scientific and Technical, 1986. Zbl 0657.16017, MR 0880019 Reference: [12] S. H. Rim, M. L. Teply: On coverings of modules.Tsukuba J. Math. 24 (2000), 15–20. MR 1791327, 10.21099/tkbjm/1496164042 Reference: [13] M. L. Teply: Torsion-free covers II.Israel J. Math. 23 (1976), 132–136. Zbl 0321.16014, MR 0417245 Reference: [14] M. L. Teply: Some aspects of Goldie’s torsion theory.Pacif. J. Math. 29 (1969), 447–459. Zbl 0174.06803, MR 0244323, 10.2140/pjm.1969.29.447 Reference: [15] J. Xu: Flat Covers of Modules.Lect. Notes Math. 1634, Springer, Berlin, 1996. Zbl 0860.16002, MR 1438789 .

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