Differential simplicity and complete integral closure

Abstract

Let gf be any set of derivations of R. Seidenberg has shown that the conductor C = {x e xRf c R) is a £^-ideal of R, so that when is ^-simple and C Φ 0, then = Rf. We investigate here the situation when C — 0. The first observation that one must make is that it is no longer true that = Rf when is differentiably simple, even when is Noetherian. We show this in Example 2.2 where we construct a 1dimensional local domain containing the rational numbers which is differentiably simple but not integrally closed. This counterexamples a conjecture of Posner [4, p. 1421] and also answers affirmatively a question of Vasconcelos [6, p. 230]. Thus, it is not a redundant task to study the relationship between a differentiably simple ring and its complete integral closure. An important tool in this study is the technique of § 3 which associates to any prime ideal P of containing no ZMdeal a rank-1, discrete valuation ring centered on P; by means of this, we show in Theorem 3.2 that over such a prime ideal P of there lies a unique prime ideal of R When is a Noetherian ϋ^-simple ring with {Pa}aeΛ as set of minimal prime ideals, Theorem 3.3 asserts that R' = f\aeA {Ra Ra is the valuation ring associated with the minimal prime ideal Pa}; Corollary 3.5 asserts that Rf is the largest ^-simple overring of having a prime ideal lying over every minimal prime ideal of R. 1* Preliminaries. Our notation and terminology adhere to that of Zariski-Samue l [7] and [8]. Throughout the paper we use to denote a commutative ring with 1, K to denote the total quotient ring of R, and A to denote an ideal of R; A is proper if A Φ R. A derivation D of is a map of into such that