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The transcendence degree of an integral domain over a subfield and the dimension of the domain. (English) Zbl 1079.13006

Let \(A\) be an integral domain of finite transcendence degree over a subfield \(k\). The author defines td\(_kA\), the transcendence degree of \(A\) with respect to \(k\), as the minimum of the transcendence degrees of \(A\) over subfields containing \(k\). Using this dimension the author obtains criteria for catenarity and universal catenarity of \(A\), if \(A\) is Noetherian and \(\dim A_M=\text{td}_kA_M\) for any maximal ideal \(M\). Further results are obtained, some of them generalizing known theorems on affine domains.

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13E05 Commutative Noetherian rings and modules
13B30 Rings of fractions and localization for commutative rings
12F20 Transcendental field extensions
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