| Title:
|
Relative exact covers (English) |
| Author:
|
Bican, Ladislav |
| Author:
|
Torrecillas, Blas |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
601-607 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Recently Rim and Teply [11] found a necessary condition for the existence of $\sigma$-torsionfree covers with respect to a given hereditary torsion theory for the category $R$-mod. This condition uses the class of $\sigma$-exact modules; i.e. the $\sigma$-torsionfree modules for which every its $\sigma$-torsionfree homomorphic image is $\sigma$-injective. In this note we shall show that the existence of $\sigma$-torsionfree covers implies the existence of $\sigma$-exact covers, and we shall investigate some sufficient conditions for the converse. (English) |
| Keyword:
|
precover |
| Keyword:
|
cover |
| Keyword:
|
hereditary torsion theory $\sigma $ |
| Keyword:
|
$\sigma $-injective module |
| Keyword:
|
$\sigma $-exact module |
| Keyword:
|
$\sigma $-pure submodule |
| MSC:
|
16D50 |
| MSC:
|
16D90 |
| MSC:
|
16S90 |
| MSC:
|
18E40 |
| idZBL:
|
Zbl 1068.16039 |
| idMR:
|
MR1883369 |
| . |
| Date available:
|
2009-01-08T19:16:43Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119276 |
| . |
| Reference:
|
[1] Anderson F.W., Fuller K.R.: Rings and Categories of Modules.Graduate Texts in Mathematics, vol.13 Springer-Verlag (1974). Zbl 0301.16001, MR 0417223 |
| Reference:
|
[2] Bican L., El Bashir R., Enochs E.: All modules have flat covers.Bull. London Math. Soc. 33 (2001), 385-390. Zbl 1029.16002, MR 1832549 |
| Reference:
|
[3] Bican L., Kepka T., Němec P.: Rings, Modules, and Preradicals.Marcel Dekker New York (1982). MR 0655412 |
| Reference:
|
[4] Bican L., Torrecillas B.: On covers.J. Algebra 236 (2001), 645-650. Zbl 0973.16002, MR 1813494 |
| Reference:
|
[5] Bican L., Torrecillas B.: Precovers.to appear. Zbl 1016.16003, MR 1962008 |
| Reference:
|
[6] Bican L., Torrecillas B.: On the existence of relative injective covers.to appear. Zbl 1006.16006, MR 1905180 |
| Reference:
|
[7] Enochs E.: Injective and flat covers, envelopes and resolvents.Israel J. Math. 39 (1981), 189-209. Zbl 0464.16019, MR 0636889 |
| Reference:
|
[8] García Rozas J.R., Torrecillas B.: On the existence of covers by injective modules relative to a torsion theory.Comm. Algebra 24 (1996), 1737-1748. MR 1386494 |
| Reference:
|
[9] Golan J.: Torsion Theories.Pitman Monographs and Surveys in Pure an Applied Mathematics, 29 Longman Scientific and Technical (1986). Zbl 0657.16017, MR 0880019 |
| Reference:
|
[10] Rada J., Saorín M.: Rings characterized by (pre)envelopes and (pre)covers of their modules.Comm. Algebra 26 (1998), 899-912. Zbl 0908.16003, MR 1606190 |
| Reference:
|
[11] Rim S.H., Teply M.L.: On coverings of modules.to appear. Zbl 0985.16017, MR 1791327 |
| Reference:
|
[12] Teply M.: Torsion-free covers II.Israel J. Math. 23 (1976), 132-136. Zbl 0321.16014, MR 0417245 |
| Reference:
|
[13] Torrecillas B.: T-torsionfree T-injective covers.Comm. Algebra 12 (1984), 2707-2726. MR 0757788 |
| Reference:
|
[14] Xu J.: Flat covers of modules.Lecture Notes in Mathematics, 1634, Springer Verlag Berlin-Heidelberg-New York (1996). Zbl 0860.16002, MR 1438789 |
| . |
