Geodesic
Non-singular covers over monoid rings
Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 9-17

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2008.133940
Classification : 06F05, 16D50, 16D80, 16S36, 16S90, 18E40
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 
Bican, Ladislav. Non-singular covers over monoid rings. Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 9-17. doi: 10.21136/MB.2008.133940
@article{10_21136_MB_2008_133940,
     author = {Bican, Ladislav},
     title = {Non-singular covers over monoid rings},
     journal = {Mathematica Bohemica},
     pages = {9--17},
     year = {2008},
     volume = {133},
     number = {1},
     doi = {10.21136/MB.2008.133940},
     mrnumber = {2400147},
     zbl = {1170.16022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133940/}
}
TY  - JOUR
AU  - Bican, Ladislav
TI  - Non-singular covers over monoid rings
JO  - Mathematica Bohemica
PY  - 2008
SP  - 9
EP  - 17
VL  - 133
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133940/
DO  - 10.21136/MB.2008.133940
LA  - en
ID  - 10_21136_MB_2008_133940
ER  - 
%0 Journal Article
%A Bican, Ladislav
%T Non-singular covers over monoid rings
%J Mathematica Bohemica
%D 2008
%P 9-17
%V 133
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133940/
%R 10.21136/MB.2008.133940
%G en
%F 10_21136_MB_2008_133940

[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, 1974. | MR

[2] L. Bican: 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. | MR | Zbl

[3] L. Bican: Precovers and Goldie’s torsion theory. Math. Bohem. 128 (2003), 395–400. | MR | Zbl

[4] L. Bican: On torsionfree classes which are not precover classes. (to appear). | MR | Zbl

[5] L. Bican: Non-singular precovers over polynomial rings. Comment. Math. Univ. Carol. 47 (2006), 369–377. | MR | Zbl

[6] L. Bican: Non-singular covers over ordered monoid rings. Math. Bohem. 131 (2006), 95–104. | MR | Zbl

[7] L. Bican, R. El Bashir, E. Enochs: All modules have flat covers. Proc. London Math. Society 33 (2001), 385–390. | MR

[8] L. Bican, B. Torrecillas: Precovers. Czech. Math. J. 53 (2003), 191–203. | MR

[9] L. Bican, B. Torrecillas: On covers. J. Algebra 236 (2001), 645–650. | DOI | MR

[10] L. Bican, T. Kepka, P. Němec: Rings, Modules, and Preradicals. Marcel Dekker, New York, 1982. | MR

[11] J. Golan: Torsion Theories. Pitman Monographs and Surveys in Pure an Applied Mathematics 29, Longman Scientific and Technical, 1986. | MR | Zbl

[12] S. H. Rim, M. L. Teply: On coverings of modules. Tsukuba J. Math. 24 (2000), 15–20. | DOI | MR

[13] M. L. Teply: Torsion-free covers II. Israel J. Math. 23 (1976), 132–136. | MR | Zbl

[14] M. L. Teply: Some aspects of Goldie’s torsion theory. Pacif. J. Math. 29 (1969), 447–459. | DOI | MR | Zbl

[15] J. Xu: Flat Covers of Modules. Lect. Notes Math. 1634, Springer, Berlin, 1996. | MR | Zbl

Cité par Sources :