Non-singular precovers over polynomial rings

Ladislav Bican

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 369-377
  • ISSN: 0010-2628

Abstract

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One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory τ for the category R -mod with τ ≥ σ , σ being Goldie’s torsion theory, the class of all τ -torsionfree modules forms a (pre)cover class if and only if τ is of finite type. The purpose of this note is to show that all members of the countable set 𝔐 = { R , R / σ ( R ) , R [ x 1 , ⋯ , x n ] , R [ x 1 , ⋯ , x n ] / σ ( R [ x 1 , ⋯ , x n ] ) , n < ω } of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.

How to cite

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Bican, Ladislav. "Non-singular precovers over polynomial rings." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 369-377. <http://eudml.org/doc/249856>.

@article{Bican2006,
abstract = {One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory $\tau $ for the category $R$-mod with $\tau \ge \sigma $, $\sigma $ being Goldie’s torsion theory, the class of all $\tau $-torsionfree modules forms a (pre)cover class if and only if $\tau $ is of finite type. The purpose of this note is to show that all members of the countable set $\mathfrak \{M\} = \lbrace R, R/\sigma (R), R[x_1,\dots ,x_n], R[x_1,\dots ,x_n]/\sigma (R[x_1,\dots ,x_n]), n <\omega \rbrace $ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.},
author = {Bican, Ladislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {hereditary torsion theory; torsion theory of finite type; Goldie's torsion theory; non-singular module; non-singular ring; precover class; cover class; hereditary torsion theories; torsion theories of finite type; Goldie torsion theory; non-singular modules; non-singular rings; precover classes; cover classes},
language = {eng},
number = {3},
pages = {369-377},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Non-singular precovers over polynomial rings},
url = {http://eudml.org/doc/249856},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Bican, Ladislav
TI - Non-singular precovers over polynomial rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 369
EP - 377
AB - One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory $\tau $ for the category $R$-mod with $\tau \ge \sigma $, $\sigma $ being Goldie’s torsion theory, the class of all $\tau $-torsionfree modules forms a (pre)cover class if and only if $\tau $ is of finite type. The purpose of this note is to show that all members of the countable set $\mathfrak {M} = \lbrace R, R/\sigma (R), R[x_1,\dots ,x_n], R[x_1,\dots ,x_n]/\sigma (R[x_1,\dots ,x_n]), n <\omega \rbrace $ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an arbitrary member of this set.
LA - eng
KW - hereditary torsion theory; torsion theory of finite type; Goldie's torsion theory; non-singular module; non-singular ring; precover class; cover class; hereditary torsion theories; torsion theories of finite type; Goldie torsion theory; non-singular modules; non-singular rings; precover classes; cover classes
UR - http://eudml.org/doc/249856
ER -

References

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  3. Bican L., Precovers and Goldie's torsion theory, Math. Bohemica 128 (2003), 395-400. (2003) Zbl1057.16027MR2032476
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  7. Bican L., Torrecillas B., On covers, J. Algebra 236 (2001), 645-650. (2001) Zbl0973.16002MR1813494
  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 and Applied Mathematics, 29 Longman Scientific and Technical, Harlow (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. Teply M.L., Torsion-free covers II, Israel J. Math. 23 (1976), 132-136. (1976) Zbl0321.16014MR0417245
  12. Teply M.L., Some aspects of Goldie's torsion theory, Pacific J. Math. 29 (1969), 447-459. (1969) Zbl0174.06803MR0244323
  13. Xu J., Flat Covers of Modules, Lecture Notes in Mathematics 1634 Springer, Berlin-Heidelberg-New York (1996). (1996) Zbl0860.16002MR1438789

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