| Title:
|
$f$-derivations on rings and modules (English) |
| Author:
|
Bland, Paul E. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
379-390 |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $\tau $ is a hereditary torsion theory on $\bold{Mod}_{R}$ and $Q_{\tau }:\bold{Mod}_{R}\rightarrow \bold{Mod}_{R}$ is the localization functor, then we show that every $f$-derivation $d:M\rightarrow N$ has a unique extension to an $f_{\tau }$-derivation $d_{\tau }:Q_{\tau }(M)\rightarrow Q_{\tau }(N)$ when $\tau $ is a differential torsion theory on $\bold{Mod}_{R}$. Dually, it is shown that if $\tau $ is cohereditary and $C_{\tau }:\bold{Mod}_{R}\rightarrow \bold{Mod}_{R}$ is the colocalization functor, then every $f$-derivation $d:M\rightarrow N$ can be lifted uniquely to an $f_{\tau }$-derivation $d_{\tau }:C_{\tau }(M)\rightarrow C_{\tau }(N)$. (English) |
| Keyword:
|
torsion theory |
| Keyword:
|
differential filter |
| Keyword:
|
localization |
| Keyword:
|
colocalization |
| Keyword:
|
$f$-derivation |
| MSC:
|
16D99 |
| MSC:
|
16S90 |
| MSC:
|
16W25 |
| idZBL:
|
Zbl 1106.16038 |
| idMR:
|
MR2281001 |
| . |
| Date available:
|
2009-05-05T16:58:03Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119600 |
| . |
| Reference:
|
[1] Alin J.S., Armendariz E.P.: TTF classes over perfect rings.J. Austral. Math. Soc. 11 (1970), 499-503. Zbl 0221.16006, MR 0274495 |
| Reference:
|
[2] Anderson F.W., Fuller K.R.: Rings and Categories of Modules.Springer, Berlin, 1973. Zbl 0765.16001, MR 1245487 |
| Reference:
|
[3] Beachy J.A.: Cotorsion radical and projective modules.Bull. Austral Math. Soc. 5 (1971), 241-253. MR 0292879 |
| Reference:
|
[4] Bland P.E.: Differential torsion theory.J. Pure Appl. Algebra 204 (2006), 1 1-8. Zbl 1102.16020, MR 2183307 |
| Reference:
|
[5] Bland P.E.: Topics in Torsion Theory.Mathematical Research 103, Wiley-VCH, Berlin, 1998. Zbl 0899.16013, MR 1640903 |
| Reference:
|
[6] Dlab V.: A characterization of perfect rings.Pacific J. Math. 33 (1970), 79-88. Zbl 0209.07201, MR 0262297 |
| Reference:
|
[7] Gabriel P.: Des catégories abeliennes.Bull. Soc. Math. France 90 (1962), 323-448. Zbl 0201.35602, MR 0232821 |
| Reference:
|
[8] Bronn S.D.: Cotorsion theories.Pacific J. Math. 48 (1973), 355-363. Zbl 0285.16026, MR 0393131 |
| Reference:
|
[9] Golan J.S.: Extension of derivations to modules of quotients.Comm. Algebra 9 3 (1981), 275-281. Zbl 0454.16020, MR 0603349 |
| Reference:
|
[10] Golan J.S.: Torsion Theories.Pitman Monographs and Surveys in Pure and Applied Mathematics 29, Longman Scientific and Technical, Harlow, 1986. Zbl 0695.16021, MR 0880019 |
| Reference:
|
[11] Jans J.P.: Some aspects of torsion.Pacific J. Math. 15 (1965), 1249-1259. Zbl 0142.28002, MR 0191936 |
| Reference:
|
[12] Lam T.Y.: Lecture on Modules and Rings.Graduate Texts in Mathematics 189, Springer, New York, 1999. |
| Reference:
|
[13] McMaster R.J.: Cotorsion theories and colocalization.Canad. J. Math. 27 (1971), 618-628. MR 0401818 |
| Reference:
|
[14] Ohtake K.: Colocalization and localization.J. Pure Appl. Algebra 11 (1977), 217-241. MR 0470027 |
| Reference:
|
[15] Sato M.: The concrete description of colocalization.Proc. Japan Acad. 52 (1976), 501-504. MR 0432688 |
| . |
