Minimal Overrings and Intermediate Rings
by
A. Jaballah
University of Waikato

Let R be an integral domain, and let K be its field of fractions. A ring S that contains R and is contained in K is called an overring of R. We denote with [R, S] the set of intermediate rings, ie rings T such that R subset or equal T subset or equal S. Some estimations for the length of chains and for the number of rings in [R, S] are known. The main purpose of this work is to give much more precise estimations for the same problems.

\Phi will denote the usual contraction map from Spec(S) to Spec(R), ie \Phi(Q) = Q \cap R for each prime ideal Q of S. We suppose that (R, S) is a normal pair, ie each ring T in [R, S] is integrally closed in S. If R is a valuation or a Prüfer domain, then (R, K) is a normal pair.

We say that the ring extension R subset S is a minimal integrally closed ring extension if R is integrally closed in S and if there is no ring T such that R subset T subset S. The main idea in the approximations provided in this work is to measure the length of maximal chains of intermediary rings. If R=R₀ subset R₁ subset ... subset R_n = S is such a chain, then each subextension R_i subset R_i+1, i=0, ..., n-1 is a minimal integrally closed ring extension. We start giving the following characterization of such subextensions:

R subset S is a minimal integrally closed ring extension if and only if (R, S) is a normal pair and |Spec(R)\\Phi(Spec(S))|=1.

Then we formulate the exact length of maximal chains of intermediate rings in normal pairs, and we deduce the length of maximal chains of overrings of Prüfer domains.

Finally we deduce an approximation for the number of intermediate rings. In particular if R is a Prüfer domain of finite spectrum, then:

|Spec(R)| <= |[R, K]| <= (dim(R)+1)|Max(R)|.

Date received: June 14, 1998