Completion of a Globalized Pseudo-valuation Domain

Let R be a pseudo-valuation domain with associated valuation domain V and I a nonzero proper ideal of R. Let R̂ (resp., V̂ ) be the I-adic (resp., IV -adic) completion of R (resp., V ). We show that R̂ is a pseudo-valuation domain (which may be a field); and that if I 6= I2, then V̂ is the associated valuation domain of R̂. Let R be an SFT globalized pseudo-valuation domain with associated Prüfer domain T and I a nonzero proper ideal of R. Let R̂ (resp., T̂ ) be the I-adic (resp., IT -adic) completion of R (resp., T ). We also show that R̂ is an SFT globalized pseudo-valuation ring with associated Prüfer ring T̂ ; and that R̂ is an SFT globalized pseudo-valuation domain if and only if √ I is a prime ideal. In [?], we proved that if V is a valuation domain and I is a nonzero proper ideal of V , then the I-adic completion V̂ of V is also a valuation domain ; if T is an SFT Prüfer domain, I a nonzero proper ideal of T , and T̂ the I-adic completion of T , then (1) T̂ is an SFT Prüfer ring, (2) T̂ is an SFT Prüfer domain if and only if √ I is a prime ideal. In this paper, we generalize these to a pseudo-valuation domain and an SFT globalized pseudo-valuation domain. First, recall from [?] that an integral domain R is said to be a pseudo-valuation domain (or, in short, a PVD) in case R has a valuation overring V such that Spec(R) = Spec(V ) as sets. For such a ring R, V is uniquely determined and is called the associated valuation domain of R, and R is quasilocal. Next, recall from [?] that an ideal I of a commutative ring R is an SFT-ideal if there exists a positive integer k and a finitely generated ideal J such that J ⊆ I and a ∈ J for each a ∈ I ; moreover, if each ideal of R is an SFT-ideal, then we say that R is an SFT-ring. Finally, recall from [?, Theorem 3.1] that each domain R for which there exists a Prüfer domain T satisfying the following two conditions is called a globalized pseudo-valuation domain (or, in short, a GPVD); and T is called the Prüfer domain associated to R : (a) R ⊆ T is a unibranched extension ; (b) There exists a nonzero radical ideal A common to T and R such that each prime ideal of T (resp., R) which contains A is a maximal ideal of T (resp., R). We start with a pullback characterization of the PVDs. Proposition 1. [?, Proposition 2.6] PVDs are precisely the pullbacks in the category of commutative rings with identity of diagrams of the form