| Title:
|
On a generalization of $QI$-rings (English) |
| Author:
|
Jirásko, J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
3 |
| Year:
|
1999 |
| Pages:
|
441-446 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L'opez-Permouth in [12]. A characterization of the $QI$-property in the category $\sigma[M]$ then follows as a consequence. (English) |
| Keyword:
|
$s$-$QI$-rings |
| Keyword:
|
$s$-stable preradicals |
| Keyword:
|
weakly $s$-divisible modules |
| Keyword:
|
$s$-tight modules |
| MSC:
|
16D50 |
| MSC:
|
16N80 |
| MSC:
|
16S90 |
| idZBL:
|
Zbl 1014.16003 |
| idMR:
|
MR1732491 |
| . |
| Date available:
|
2009-01-08T18:53:51Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119100 |
| . |
| Reference:
|
[1] Ahsan J., Enochs E.: Rings all of whose torsion quasi-injective modules are injective.Glasgow Math. J. 25 (1984), 219-227. Zbl 0541.16019, MR 0752817 |
| Reference:
|
[2] Al-Huzali, Jain S.K., López-Permouth S.R.: On the weak relative injectivity of rings and modules.Lecture Notes in Math. 1448, Springer Verlag, 1990, pp.93-98. MR 1084625 |
| Reference:
|
[3] Azumaya G., Mbuntum F., Varadarajan K.: On $M$-projective and $M$-injective modules.Pacific J. Math. 59 (1975), 9-16. Zbl 0316.16027, MR 0384867 |
| Reference:
|
[4] Beachy J.A.: A generalization of injectivity.Pacific J. Math. 41 (1972), 313-328. Zbl 0235.16018, MR 0313317 |
| Reference:
|
[5] Bican L., Kepka T., Němec P.: Rings, Modules and Preradicals.Marcel Dekker Inc., New York and Basel, 1982. MR 0655412 |
| Reference:
|
[6] Boyle A.K.: Hereditary QI-rings.Trans. Amer. Math. Soc. 192 (1974), 115-120. Zbl 0286.16016, MR 0338075 |
| Reference:
|
[7] Bunu I.D.: On left QI-rings (in Russian).Mat. Issled. 49 (1979), 23-34. MR 0544974 |
| Reference:
|
[8] Byrd K.A.: When are quasi-injectives injective?.Canad. Math. Bull. 15 (1972), 599-600. Zbl 0248.16016, MR 0311715 |
| Reference:
|
[9] Byrd K.A.: Rings, whose quasi-injective modules are injective.Proc. Amer. Math. Soc. 33 (1972), 235-240. Zbl 0214.05605, MR 0310009 |
| Reference:
|
[10] Gabriel P.: Des catégories abéliennes.Bull. Soc. Math. France 90 (1962), 323-448. Zbl 0201.35602, MR 0232821 |
| Reference:
|
[11] Golan J.S.: Torsion Theories.Longman Scientific and Technical, Harlow, 1986. Zbl 0695.16021, MR 0880019 |
| Reference:
|
[12] Golan J.S., López-Permouth S.R.: QI-filters and tight modules.Comm. Algebra 19 (8) (1991), 2217-2229. MR 1123120 |
| Reference:
|
[13] Jain S.K., López-Permouth S.R., Singh S.: On a class of QI-rings.Glasgow Math. J. 34 (1992), 75-81. MR 1145633 |
| Reference:
|
[14] Jirásko J.: Generalized injectivity.Comment. Math. Univ. Carolinae 16 (1975), 621-636. MR 0399165 |
| Reference:
|
[15] Morimoto S.: Weakly divisible and divisible modules.Tsukuba J. Math. 6 (2) (1982), 195-200. Zbl 0529.16019, MR 0705113 |
| . |
