Precovers and Goldie’s torsion theory

Ladislav Bican

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 4, page 395-400
  • ISSN: 0862-7959

Abstract

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Recently, Rim and Teply , using the notion of τ -exact modules, found a necessary condition for the existence of τ -torsionfree covers with respect to a given hereditary torsion theory τ for the category R -mod of all unitary left R -modules over an associative ring R with identity. Some relations between τ -torsionfree and τ -exact covers have been investigated in . The purpose of this note is to show that if σ = ( 𝒯 σ , σ ) is Goldie’s torsion theory and σ is a precover class, then τ is a precover class whenever τ σ . Further, it is shown that σ is a cover class if and only if σ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that τ is a cover class for all hereditary torsion theories τ σ .

How to cite

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Bican, Ladislav. "Precovers and Goldie’s torsion theory." Mathematica Bohemica 128.4 (2003): 395-400. <http://eudml.org/doc/249217>.

@article{Bican2003,
abstract = {Recently, Rim and Teply , using the notion of $\tau $-exact modules, found a necessary condition for the existence of $\tau $-torsionfree covers with respect to a given hereditary torsion theory $\tau $ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau $-torsionfree and $\tau $-exact covers have been investigated in . The purpose of this note is to show that if $\sigma = (\mathcal \{T\}_\{\sigma \},\mathcal \{F\}_\{\sigma \})$ is Goldie’s torsion theory and $\mathcal \{F\}_\{\sigma \}$ is a precover class, then $\mathcal \{F\}_\{\tau \}$ is a precover class whenever $\tau \ge \sigma $. Further, it is shown that $\mathcal \{F\}_\{\sigma \}$ is a cover class if and only if $\sigma $ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that $\mathcal \{F\}_\{\tau \}$ is a cover class for all hereditary torsion theories $\tau \ge \sigma $.},
author = {Bican, Ladislav},
journal = {Mathematica Bohemica},
keywords = {hereditary torsion theory; Goldie’s torsion theory; non-singular ring; precover class; cover class; torsionfree covers; lattices of torsion theories; hereditary torsion theories; Goldie torsion theory; non-singular rings; precover classes; cover classes; torsionfree covers; lattices of torsion theories},
language = {eng},
number = {4},
pages = {395-400},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Precovers and Goldie’s torsion theory},
url = {http://eudml.org/doc/249217},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Bican, Ladislav
TI - Precovers and Goldie’s torsion theory
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 4
SP - 395
EP - 400
AB - Recently, Rim and Teply , using the notion of $\tau $-exact modules, found a necessary condition for the existence of $\tau $-torsionfree covers with respect to a given hereditary torsion theory $\tau $ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau $-torsionfree and $\tau $-exact covers have been investigated in . The purpose of this note is to show that if $\sigma = (\mathcal {T}_{\sigma },\mathcal {F}_{\sigma })$ is Goldie’s torsion theory and $\mathcal {F}_{\sigma }$ is a precover class, then $\mathcal {F}_{\tau }$ is a precover class whenever $\tau \ge \sigma $. Further, it is shown that $\mathcal {F}_{\sigma }$ is a cover class if and only if $\sigma $ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that $\mathcal {F}_{\tau }$ is a cover class for all hereditary torsion theories $\tau \ge \sigma $.
LA - eng
KW - hereditary torsion theory; Goldie’s torsion theory; non-singular ring; precover class; cover class; torsionfree covers; lattices of torsion theories; hereditary torsion theories; Goldie torsion theory; non-singular rings; precover classes; cover classes; torsionfree covers; lattices of torsion theories
UR - http://eudml.org/doc/249217
ER -

References

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  2. Precovers, Czechoslovak Math. J. 53 (2003), 191–203. (2003) MR1962008
  3. 10.1006/jabr.2000.8562, J. Algebra 236 (2001), 645–650. (2001) MR1813494DOI10.1006/jabr.2000.8562
  4. All modules have flat covers, Proc. London Math. Society 33 (2001), 385–390. (2001) MR1832549
  5. Relative exact covers, Comment. Math. Univ. Carolinae 42 (2001), 601–607. (2001) MR1883369
  6. Rings, Modules, and Preradicals, Marcel Dekker, New York, 1982. (1982) MR0655412
  7. Torsion Theories, Pitman Monographs and Surveys in Pure an Applied Matematics, 29, Longman Scientific and Technical, 1986. (1986) Zbl0657.16017MR0880019
  8. On coverings of modules, Tsukuba J. Math. 24 (2000), 15–20. (2000) MR1791327
  9. Torsion-free covers II, Israel J. Math. 23 (1976), 132–136. (1976) Zbl0321.16014MR0417245
  10. Flat Covers of Modules, Lecture Notes in Mathematics 1634, Springer, Berlin, 1996. (1996) Zbl0860.16002MR1438789

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