Notes on Local Integral Extension Domains

Abstract

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3]. In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).