COMPLETION OF A GLOBALIZED PSEUDO-VALUATION DOMAIN

Abstract

Let R be a pseudo-valuation domain with associated valuation domain V and I a nonzero proper ideal of R. Let (R) over cap R (resp., (V) over cap) be the I-adic (resp., IV-adic) completion of R (resp., V). We show that (R) over cap is a pseudo-valuation domain (which may be a field); and that if I not equal I-2, then (V) over cap is the associated valuation domain of (R) over cap. Let R be an SFT globalized pseudo-valuation domain with associated Prufer domain T and I a nonzero proper ideal of R. Let (R) over cap (resp., (T) over cap) be the I-adic (resp., IT-adic) completion of R (resp., T). We also show that (R) over cap is an SFT globalized pseudo-valuation ring with associated Prufer ring (T) over cap; and that (R) over cap is an SFT globalized pseudo-valuation domain if and only if rootI is a prime ideal.