×

zbMATH — the first resource for mathematics

Precovers. (English) Zbl 1016.16003
Summary: Let \(\mathcal G\) be an abstract class (closed under isomorphic copies) of left \(R\)-modules. In the first part of the paper some sufficient conditions under which \(\mathcal G\) is a precover class are given. The next section studies the \(\mathcal G\)-precovers which are \(\mathcal G\)-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category of left \(R\)-modules. Especially, several sufficient conditions for the existence of \(\sigma\)-torsionfree and \(\sigma\)-torsionfree \(\sigma\)-injective covers are presented.

MSC:
16D90 Module categories in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] L. Bican, T. Kepka and P. Němec: Rings, Modules, and Preradicals. Marcel Dekker, New York, 1982. · Zbl 0483.16026
[2] E. Enochs: Torsion free covering modules II. Proc. Amer. Math. Soc. 114 (1963), 884-889. · Zbl 0116.26003
[3] J. Golan: Torsion Theories. Pitman Monographs and Surveys in Pure an Applied Matematics, vol. 29, Longman Scientific and Technical, 1986. · Zbl 0657.16017
[4] J. R. García Rozas and B. Torrecillas: On the existence of covers by injective modules relative to a torsion theory. Comm. Algebra 24 (1996), 1737-1748. · Zbl 0849.16034
[5] J. Rada and M. Saorín: Rings characterized by (pre)envelopes and (pre)covers of their modules. Comm. Algebra 26 (1998), 899-912. · Zbl 0908.16003
[6] M. Teply: Torsionfree injective modules. Pacific J. Math. 28 (1969), 441-453. · Zbl 0172.32001
[7] M. Teply: Torsion-free covers II. Israel J. Math. 23 (1976), 132-136. · Zbl 0321.16014
[8] B. Torrecillas: T-torsionfree T-injective covers. Comm. Algebra 12 (1984), 2707-2726. · Zbl 0542.16025
[9] J. Xu: Flat Covers of Modules. Lecture Notes in Mathematics, 1634 Springer Verlag, Berlin-Heidelberg-New York, 1996. · Zbl 0860.16002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.