Previous |  Up |  Next

Article

Keywords:
hereditary torsion theory; exact; noetherian and perfect torsion theory; Goldie’s torsion theory; precover class; cover class; precover and cover of a module
Summary:
In the class of all exact torsion theories the torsionfree classes are cover (precover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented.
References:
[1] F. W. Anderson and K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics, Springer-Verlag, 1974. MR 0417223
[2] L. Bican: Torsionfree precovers. Contributions to General Algebra 15, Proceedings of the Klagenfurt Conference 2003 (AAA 66), Verlag Johannes Heyn, Klagenfurt 2004 15 (2004), 1–6. MR 2080845 | Zbl 1074.16002
[3] L. Bican: Relatively exact modules. Comment. Math. Univ. Carolinae 44 (2003), 569–574. MR 2062873 | Zbl 1101.16023
[4] L. Bican: Precovers and Goldie’s torsion theory. Math. Bohem. 128 (2003), 395–400. MR 2032476 | Zbl 1057.16027
[5] L. Bican: On precover classes. Ann. Univ. Ferrara Sez. VII Sc. Mat. LI (2005), 61–67. MR 2294759 | Zbl 1122.16001
[6] L. Bican, R. El Bashir and E. Enochs: All modules have flat covers. Proc. London Math. Society 33 (2001), 649–652. MR 1832549
[7] L. Bican and B. Torrecillas: Precovers. Czech. Math. J. 53 (2003), 191–203. MR 1962008
[8] L. Bican and B. Torrecillas: On covers. J. Algebra 236 (2001), 645–650. DOI 10.1006/jabr.2000.8562 | MR 1813494
[9] L. Bican and B. Torrecillas: On the existence of relative injective covers. Acta Math. Hungar. 95 (2002), 178–186. MR 1905180
[10] L. Bican and B. Torrecillas: Relative exact covers. Comment. Math. Univ. Carolinae 42 (2001), 477–487. MR 1883369
[11] L. Bican, T. Kepka and P. Němec: Rings, Modules, and Preradicals. Marcel Dekker, New York, 1982. MR 0655412
[12] J. Golan: Torsion Theories. Pitman Monographs and Surveys in Pure an Applied Matematics, 29, Longman Scientific and Technical, 1986. MR 0880019 | Zbl 0657.16017
[13] J. R. García Rozas and B. Torrecillas: On the existence of covers by injective modules relative to a torsion theory. Comm. Alg. 24 (1996), 1737–1748. DOI 10.1080/00927879608825667 | MR 1386494
[14] S. H. Rim and M. L. Teply: On coverings of modules. Tsukuba J. Math. 24 (2000), 15–20. MR 1791327
[15] M. L. Teply: Torsion-free covers II. Israel J. Math. 23 (1976), 132–136. MR 0417245 | Zbl 0321.16014
[16] M. L. Teply: Some aspects of Goldie’s torsion theory. Pacif. J. Math. 29 (1969), 447–459. DOI 10.2140/pjm.1969.29.447 | MR 0244323 | Zbl 0174.06803
[17] J. Xu: Flat Covers of Modules. Lecture Notes in Mathematics 1634, Springer Verlag Berlin-Heidelberg-New York, 1996. MR 1438789 | Zbl 0860.16002
Partner of
EuDML logo