Non-singular covers over monoid rings

Ladislav Bican

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 1, page 9-17
  • ISSN: 0862-7959

Abstract

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We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If G is a strongly cancellative monoid such that h G ⊆ G h for each h ∈ G and if R is a ring such that a R ⊆ R a for each a ∈ R , then the class of all non-singular left R -modules is a cover class if and only if the class of all non-singular left R G -modules is a cover class. These two conditions are also equivalent whenever we replace the strongly cancellative monoid G by the totally ordered cancellative monoid or by the totally ordered group.

How to cite

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Bican, Ladislav. "Non-singular covers over monoid rings." Mathematica Bohemica 133.1 (2008): 9-17. <http://eudml.org/doc/250515>.

@article{Bican2008,
abstract = {We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever we replace the strongly cancellative monoid $G$ by the totally ordered cancellative monoid or by the totally ordered group.},
author = {Bican, Ladislav},
journal = {Mathematica Bohemica},
keywords = {hereditary torsion theory; torsion theory of finite type; Goldie’s torsion theory; non-singular module; non-singular ring; monoid ring; precover class; cover class; hereditary torsion theories; torsion theories of finite type; Goldie torsion theory; non-singular modules; monoid rings; strongly cancellative monoids; cover classes},
language = {eng},
number = {1},
pages = {9-17},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Non-singular covers over monoid rings},
url = {http://eudml.org/doc/250515},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Bican, Ladislav
TI - Non-singular covers over monoid rings
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 1
SP - 9
EP - 17
AB - We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever we replace the strongly cancellative monoid $G$ by the totally ordered cancellative monoid or by the totally ordered group.
LA - eng
KW - hereditary torsion theory; torsion theory of finite type; Goldie’s torsion theory; non-singular module; non-singular ring; monoid ring; precover class; cover class; hereditary torsion theories; torsion theories of finite type; Goldie torsion theory; non-singular modules; monoid rings; strongly cancellative monoids; cover classes
UR - http://eudml.org/doc/250515
ER -

References

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  1. Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer, 1974. (1974) MR1245487
  2. Torsionfree precovers, Contributions to General Algebra 15, Proceedings of the 66th Workshop on General Algebra, Klagenfurt 2003, Verlag Johannes Heyn, Klagenfurt, 2004, pp. 1–6. (2004) Zbl1074.16002MR2080845
  3. Precovers and Goldie’s torsion theory, Math. Bohem. 128 (2003), 395–400. (2003) Zbl1057.16027MR2032476
  4. On torsionfree classes which are not precover classes, (to appear). (to appear) Zbl1166.16013MR2411109
  5. Non-singular precovers over polynomial rings, Comment. Math. Univ. Carol. 47 (2006), 369–377. (2006) Zbl1106.16032MR2281000
  6. Non-singular covers over ordered monoid rings, Math. Bohem. 131 (2006), 95–104. (2006) Zbl1111.16029MR2211006
  7. All modules have flat covers, Proc. London Math. Society 33 (2001), 385–390. (2001) MR1832549
  8. Precovers, Czech. Math. J. 53 (2003), 191–203. (2003) MR1962008
  9. 10.1006/jabr.2000.8562, J. Algebra 236 (2001), 645–650. (2001) MR1813494DOI10.1006/jabr.2000.8562
  10. Rings, Modules, and Preradicals, Marcel Dekker, New York, 1982. (1982) MR0655412
  11. Torsion Theories, Pitman Monographs and Surveys in Pure an Applied Mathematics 29, Longman Scientific and Technical, 1986. (1986) Zbl0657.16017MR0880019
  12. 10.21099/tkbjm/1496164042, Tsukuba J. Math. 24 (2000), 15–20. (2000) MR1791327DOI10.21099/tkbjm/1496164042
  13. Torsion-free covers II, Israel J. Math. 23 (1976), 132–136. (1976) Zbl0321.16014MR0417245
  14. 10.2140/pjm.1969.29.447, Pacif. J. Math. 29 (1969), 447–459. (1969) Zbl0174.06803MR0244323DOI10.2140/pjm.1969.29.447
  15. Flat Covers of Modules, Lect. Notes Math. 1634, Springer, Berlin, 1996. (1996) Zbl0860.16002MR1438789

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