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Keywords:
hereditary torsion theory; torsion theory of finite type; Goldie's torsion theory; non-singular module; non-singular ring; precover class; cover class
hereditary torsion theory; torsion theory of finite type; Goldie's torsion theory; non-singular module; non-singular ring; precover class; cover class
Summary:
One of the results in my previous paper {\it On torsionfree classes which are not precover classes\/}, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \geq\sigma$, $\sigma$ being Goldie's torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $\frak M = \{R, R/\sigma (R), R[x_1,\dots ,x_n], R[x_1,\dots ,x_n]/\sigma(R[x_1,\dots ,x_n]), n <\omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.
References:
[1] Anderson F.W., Fuller K.R.: Rings and Categories of Modules. Graduate Texts in Mathematics, vol.13 Springer, New York-Heidelberg (1974). MR 0417223 | Zbl 0301.16001
[2] Bican L.: Torsionfree precovers. Contributions to General Algebra 15, Proceedings of the Klagenfurt Conference 2003 (AAA 66), Verlag Johannes Heyn, Klagenfurt, 2004, pp.1-6. MR 2080845 | Zbl 1074.16002
[3] Bican L.: Precovers and Goldie's torsion theory. Math. Bohemica 128 (2003), 395-400. MR 2032476 | Zbl 1057.16027
[4] Bican L.: On torsionfree classes which are not precover classes. preprint. MR 2411109 | Zbl 1166.16013
[5] Bican L., El Bashir R., Enochs E.: All modules have flat covers. Proc. London Math. Soc. 33 (2001), 649-652. MR 1832549 | Zbl 1029.16002
[6] Bican L., Torrecillas B.: Precovers. Czechoslovak Math. J. 53 (128) (2003), 191-203. MR 1962008 | Zbl 1016.16003
[7] Bican L., Torrecillas B.: On covers. J. Algebra 236 (2001), 645-650. MR 1813494 | Zbl 0973.16002
[8] Bican L., Kepka T., Němec P.: Rings, Modules, and Preradicals. Marcel Dekker New York (1982). MR 0655412
[9] Golan J.: Torsion Theories. Pitman Monographs and Surveys in Pure and Applied Mathematics, 29 Longman Scientific and Technical, Harlow (1986). MR 0880019 | Zbl 0657.16017
[10] Rim S.H., Teply M.L.: On coverings of modules. Tsukuba J. Math. 24 (2000), 15-20. MR 1791327 | Zbl 0985.16017
[11] Teply M.L.: Torsion-free covers II. Israel J. Math. 23 (1976), 132-136. MR 0417245 | Zbl 0321.16014
[12] Teply M.L.: Some aspects of Goldie's torsion theory. Pacific J. Math. 29 (1969), 447-459. MR 0244323 | Zbl 0174.06803
[13] Xu J.: Flat Covers of Modules. Lecture Notes in Mathematics 1634 Springer, Berlin-Heidelberg-New York (1996). MR 1438789 | Zbl 0860.16002
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