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Lying-Over Pairs of Commutative Rings

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
University of Tennessee, Knoxville, Tennessee
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(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 454 - 475
Copyright
Copyright © Canadian Mathematical Society 1981

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