Integral domain extensions with the height condition
摘要
Let R ⊂ S be an extension of integral domains. We recall that R ⊂ S satisfies the height condition if, for every prime ideal Q of S, hts(Q) = htR(Q ⋂ 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 ⊂ 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 ⊂ 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 ⊆ 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.