| 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 |
| . |
