| Title:
|
Ring extensions with some finiteness conditions on the set of intermediate rings (English) |
| Author:
|
Jaballah, Ali |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
117-124 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension $R\subseteq S$ to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions. (English) |
| Keyword:
|
integral domain |
| Keyword:
|
intermediate ring |
| Keyword:
|
overring |
| Keyword:
|
integrally closed |
| Keyword:
|
Prüfer domain |
| Keyword:
|
residually algebraic pair |
| Keyword:
|
normal pair |
| Keyword:
|
primitive extension |
| Keyword:
|
a.c.c. |
| Keyword:
|
d.c.c. |
| Keyword:
|
minimal condition |
| Keyword:
|
maximal condition |
| Keyword:
|
affine extension |
| Keyword:
|
Dilworth number |
| Keyword:
|
width of an ordered set |
| MSC:
|
13B02 |
| MSC:
|
13B22 |
| MSC:
|
13E15 |
| MSC:
|
13E99 |
| MSC:
|
13F05 |
| MSC:
|
13G05 |
| idZBL:
|
Zbl 1224.13011 |
| idMR:
|
MR2595076 |
| . |
| Date available:
|
2010-07-20T16:19:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140555 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Jaballah, A.: A lower bound for the number of intermediary rings.Commun. Algebra 27 (1999), 1307-1311. Zbl 0972.13008, MR 1669083, 10.1080/00927879908826495 |
| Reference:
|
[10] Jaballah, A.: Finiteness of the set of intermediary rings in normal pairs.Saitama Math. J. 17 (1999), 59-61. Zbl 1073.13500, MR 1740247 |
| Reference:
|
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| Reference:
|
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| . |
