| Title:
|
Ideal-theoretic characterizations of valuation and Prüfer monoids (English) |
| Author:
|
Halter-Koch, Franz |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
41-46 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains. (English) |
| Keyword:
|
valuation monoids |
| Keyword:
|
Prüfer domains |
| MSC:
|
13A15 |
| MSC:
|
13F05 |
| MSC:
|
20M12 |
| MSC:
|
20M14 |
| MSC:
|
20M25 |
| idZBL:
|
Zbl 1114.20041 |
| idMR:
|
MR2054871 |
| . |
| Date available:
|
2008-06-06T22:42:51Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107889 |
| . |
| Reference:
|
[1] Aubert K. E.: Some characterizations of valuation rings.Duke Math. J. 21 (1954), 517–525. MR 0062727 |
| Reference:
|
[2] Garcia J. M., Jaros P., Santos E.: Prüfer $*$-multiplication domains and torsion theories.Comm. Algebra 27 (1999), 1275–1295. MR 1669156 |
| Reference:
|
[3] Halter-Koch F.: Ideal Systems.Marcel Dekker 1998. Zbl 0953.13001, MR 1828371 |
| Reference:
|
[4] Halter-Koch F.,: Construction of ideal systems having nice noetherian properties.Commutative Rings in a Non-Noetherian Setting (S. T. Chapman and S. Glaz, eds.), Kluwer 2000, 271–285. MR 1858166 |
| Reference:
|
[5] Halter-Koch F.: Characterization of Prüfer multiplication monoids and domains by means of spectral module systems.Monatsh. Math. 139 (2003), 19–31. Zbl 1058.20049, MR 1981115 |
| Reference:
|
[6] Halter-Koch F.: Valuation Monoids, Defining Systems and Approximation Theorems.Semigroup Forum 55 (1997), 33–56. Zbl 0880.20047, MR 1446657 |
| . |
