Relatively exact modules

Ladislav Bican

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 569-576
  • ISSN: 0010-2628

Abstract

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Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice ℰ τ ( M ) of submodules of a torsionfree module M consisting of all submodules N of M such that M / N is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of M / N is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application we also obtain some new characterizations of perfect torsion theories.

How to cite

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Bican, Ladislav. "Relatively exact modules." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 569-576. <http://eudml.org/doc/249191>.

@article{Bican2003,
abstract = {Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice $\mathcal \{E\}_\{\tau \}(M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application we also obtain some new characterizations of perfect torsion theories.},
author = {Bican, Ladislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Hereditary torsion theory $\tau $; $\tau $-injective module; $\tau $-exact module; preradical; exact torsion theory; perfect torsion theory; injective modules; hereditary torsion theories; preradicals; exact torsion theories; perfect torsion theories; relatively exact modules; torsionfree covers; lattices of submodules; relative injective hulls},
language = {eng},
number = {4},
pages = {569-576},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Relatively exact modules},
url = {http://eudml.org/doc/249191},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Bican, Ladislav
TI - Relatively exact modules
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 569
EP - 576
AB - Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice $\mathcal {E}_{\tau }(M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application we also obtain some new characterizations of perfect torsion theories.
LA - eng
KW - Hereditary torsion theory $\tau $; $\tau $-injective module; $\tau $-exact module; preradical; exact torsion theory; perfect torsion theory; injective modules; hereditary torsion theories; preradicals; exact torsion theories; perfect torsion theories; relatively exact modules; torsionfree covers; lattices of submodules; relative injective hulls
UR - http://eudml.org/doc/249191
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., Torsionfree precovers, to appear. Zbl1074.16002MR2080845
  3. Bican L., El Bashir R., Enochs E., All modules have flat covers, Proc. London Math. Society 33 (2001), 649-652. (2001) Zbl1029.16002MR1832549
  4. Bican L., Torrecillas B., Precovers, Czechoslovak Math. J. 53 (128) (2003), 191-203. (2003) Zbl1016.16003MR1962008
  5. Bican L., Torrecillas B., On covers, J. Algebra 236 (2001), 645-650. (2001) Zbl0973.16002MR1813494
  6. Bican L., Torrecillas B., On the existence of relative exact covers, Acta Math. Hungar. 95 (2002), 178-186. (2002) MR1905180
  7. Bican L., Torrecillas B., Relative exact covers, Comment. Math. Univ. Carolinae 42 (2001), 477-487. (2001) Zbl1068.16039MR1883369
  8. Bican L., Kepka T., Němec P., Rings, Modules, and Preradicals, Marcel Dekker New York (1982). (1982) MR0655412
  9. Golan J., Torsion Theories, Pitman Monographs and Surveys in Pure an Applied Mathematics, 29 Longman Scientific and Technical (1986). (1986) Zbl0657.16017MR0880019
  10. Rim S.H., Teply M.L., On coverings of modules, Tsukuba J. Math. 24 (2000), 15-20. (2000) Zbl0985.16017MR1791327
  11. 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
  12. Teply M.L., Torsion-free covers II, Israel J. Math. 23 (1976), 132-136. (1976) Zbl0321.16014MR0417245
  13. 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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