| Title:
|
Non-singular precovers over polynomial rings (English) |
| Author:
|
Bican, Ladislav |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
369-377 |
| . |
| Category:
|
math |
| . |
| 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. (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:
|
precover class |
| Keyword:
|
cover class |
| MSC:
|
16D50 |
| MSC:
|
16D80 |
| MSC:
|
16S90 |
| MSC:
|
18E40 |
| idZBL:
|
Zbl 1106.16032 |
| idMR:
|
MR2281000 |
| . |
| Date available:
|
2009-05-05T16:57:57Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119599 |
| . |
| Reference:
|
[1] Anderson F.W., Fuller K.R.: Rings and Categories of Modules.Graduate Texts in Mathematics, vol.13 Springer, New York-Heidelberg (1974). Zbl 0301.16001, MR 0417223 |
| Reference:
|
[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. Zbl 1074.16002, MR 2080845 |
| Reference:
|
[3] Bican L.: Precovers and Goldie's torsion theory.Math. Bohemica 128 (2003), 395-400. Zbl 1057.16027, MR 2032476 |
| Reference:
|
[4] Bican L.: On torsionfree classes which are not precover classes.preprint. Zbl 1166.16013, MR 2411109 |
| Reference:
|
[5] Bican L., El Bashir R., Enochs E.: All modules have flat covers.Proc. London Math. Soc. 33 (2001), 649-652. Zbl 1029.16002, MR 1832549 |
| Reference:
|
[6] Bican L., Torrecillas B.: Precovers.Czechoslovak Math. J. 53 (128) (2003), 191-203. Zbl 1016.16003, MR 1962008 |
| Reference:
|
[7] Bican L., Torrecillas B.: On covers.J. Algebra 236 (2001), 645-650. Zbl 0973.16002, MR 1813494 |
| 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 and Applied Mathematics, 29 Longman Scientific and Technical, Harlow (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] Teply M.L.: Torsion-free covers II.Israel J. Math. 23 (1976), 132-136. Zbl 0321.16014, MR 0417245 |
| Reference:
|
[12] Teply M.L.: Some aspects of Goldie's torsion theory.Pacific J. Math. 29 (1969), 447-459. Zbl 0174.06803, MR 0244323 |
| Reference:
|
[13] Xu J.: Flat Covers of Modules.Lecture Notes in Mathematics 1634 Springer, Berlin-Heidelberg-New York (1996). Zbl 0860.16002, MR 1438789 |
| . |
