Article
| Title: | Integral domain extensions with the height condition (English) |
| Author: | Zeidi, Nabil |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 663-675 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $R \subset S$ be an extension of integral domains. We recall that $R\subset S$ satisfies the height condition if, for every prime ideal $Q$ of $S$, $ht_S(Q)=ht_R(Q\cap R)$. Several characterizations of such extensions are given. For example, we prove that if there exists a finite maximal chain of rings from $R$ to $S$, and $R$ is integrally closed in $S$, then $R\subset S$ satisfies the height condition. The second purpose is to introduce and study CH-pairs. A pair $(R,S)$ is called the CH-pair if the extension $R\subset T$ satisfies the height condition for each intermediate ring $T$ between $R$ and $S$. When $R$ is a field it is shown that the pair $(R,S)$ is a CH-pair if and only if $S$ is a field algebraic over $R$. We also establish that $(R,S)$ is a CH-pair if and only if $R\subseteq R^*$ satisfies the height condition and $(R^*,S)$ is a normal pair, where $R^*$ is the integral closure of $R$ in $S$. Further consequences are also provided. (English) |
| Keyword: | prime ideal |
| Keyword: | intermediate ring |
| Keyword: | minimal extension |
| Keyword: | normal pair of rings |
| Keyword: | height condition |
| Keyword: | going down |
| Keyword: | incomparability |
| MSC: | 13A15 |
| MSC: | 13A18 |
| MSC: | 13B02 |
| MSC: | 13B21 |
| DOI: | 10.21136/CMJ.2026.0416-25 |
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| Date available: | 2026-05-22T11:24:47Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153655 |
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