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Keywords:
prime ideal; intermediate ring; minimal extension; normal pair of rings; height condition; going down; incomparability
prime ideal; intermediate ring; minimal extension; normal pair of rings; height condition; going down; incomparability
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.
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