ON THE WEAKLY ARF (S2)-IFICATIONS OF NOETHERIAN RINGS

Abstract

The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman (and recently rediscovered), being closely explored with further developments. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We call this extension the weakly Arf $(S_2)$-ification, and develop the basic theory, including some existence theorems.