Relative exact covers

Ladislav Bican; Blas Torrecillas

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 601-607
  • ISSN: 0010-2628

Abstract

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Recently Rim and Teply [11] found a necessary condition for the existence of σ -torsionfree covers with respect to a given hereditary torsion theory for the category R -mod. This condition uses the class of σ -exact modules; i.e. the σ -torsionfree modules for which every its σ -torsionfree homomorphic image is σ -injective. In this note we shall show that the existence of σ -torsionfree covers implies the existence of σ -exact covers, and we shall investigate some sufficient conditions for the converse.

How to cite

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Bican, Ladislav, and Torrecillas, Blas. "Relative exact covers." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 601-607. <http://eudml.org/doc/248797>.

@article{Bican2001,
abstract = {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.},
author = {Bican, Ladislav, Torrecillas, Blas},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {precover; cover; hereditary torsion theory $\sigma $; $\sigma $-injective module; $\sigma $-exact module; $\sigma $-pure submodule; precovers; covers; hereditary torsion theories; injective modules; exact modules; pure submodules},
language = {eng},
number = {4},
pages = {601-607},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Relative exact covers},
url = {http://eudml.org/doc/248797},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Bican, Ladislav
AU - Torrecillas, Blas
TI - Relative exact covers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 601
EP - 607
AB - 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.
LA - eng
KW - precover; cover; hereditary torsion theory $\sigma $; $\sigma $-injective module; $\sigma $-exact module; $\sigma $-pure submodule; precovers; covers; hereditary torsion theories; injective modules; exact modules; pure submodules
UR - http://eudml.org/doc/248797
ER -

References

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  1. Anderson F.W., Fuller K.R., Rings and Categories of Modules, Graduate Texts in Mathematics, vol.13 Springer-Verlag (1974). (1974) Zbl0301.16001MR0417223
  2. Bican L., El Bashir R., Enochs E., All modules have flat covers, Bull. London Math. Soc. 33 (2001), 385-390. (2001) Zbl1029.16002MR1832549
  3. Bican L., Kepka T., Němec P., Rings, Modules, and Preradicals, Marcel Dekker New York (1982). (1982) MR0655412
  4. Bican L., Torrecillas B., On covers, J. Algebra 236 (2001), 645-650. (2001) Zbl0973.16002MR1813494
  5. Bican L., Torrecillas B., Precovers, to appear. Zbl1016.16003MR1962008
  6. Bican L., Torrecillas B., On the existence of relative injective covers, to appear. Zbl1006.16006MR1905180
  7. Enochs E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209. (1981) Zbl0464.16019MR0636889
  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. (1996) MR1386494
  9. Golan J., Torsion Theories, Pitman Monographs and Surveys in Pure an Applied Mathematics, 29 Longman Scientific and Technical (1986). (1986) Zbl0657.16017MR0880019
  10. Rada J., Saorín M., Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26 (1998), 899-912. (1998) Zbl0908.16003MR1606190
  11. Rim S.H., Teply M.L., On coverings of modules, to appear. Zbl0985.16017MR1791327
  12. Teply M., Torsion-free covers II, Israel J. Math. 23 (1976), 132-136. (1976) Zbl0321.16014MR0417245
  13. Torrecillas B., T-torsionfree T-injective covers, Comm. Algebra 12 (1984), 2707-2726. (1984) MR0757788
  14. Xu J., Flat covers of modules, Lecture Notes in Mathematics, 1634, Springer Verlag Berlin-Heidelberg-New York (1996). (1996) Zbl0860.16002MR1438789

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