SATURATED CHAINS OF INTEGRALLY CLOSED OVERRINGS

If R is an integrally closed domain and R subset of T is a minimal ring extension such that T is a Prufer domain, then R is a Prufer domain. A domain R with quotient field K is the intersection of a chain of Prufer overrings if and only if R is integrally closed and there is a saturated chain C of overrings of R going from R to K such that each ring in C\{R} is a Prufer domain. In particular, if R is a Krull domain of Krull dimension at least 2 with only countably many height 1 prime ideals, then R is a non-Prufer domain having a saturated chain of integrally closed overrings going from R to the quotient field of R.