Completion of a Prüfer Domain

Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V̂ = dimV I+1 if I is not idempotent, V̂ ∼= V I if I is idempotent, (3) dim D̂ = dimD I + 1, (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring. Throughout this paper, all rings are assumed to be commutative rings with identity. It is well known that for a Noetherian ring R and a proper ideal I of R, the Krull dimension of the I-adic completion R̂ of R equals sup{htM | M is a maximal ideal of R containing I} [?, Proposition 7.3 and p. 35]. In this paper, we will study the completion of a valuation domain and a Prüfer domain and get a similar equation for the Krull dimension of the completion. First we describe some properties of prime ideals of the power series ring V [[X]] of a valuation domain V . In [?], Arnold gave a collection of principal prime ideals of V [[X]], where V is a finite-dimensional valuation domain with the SFT-property, i.e., a finite-dimensional discrete valuation domain: Let Q be a prime ideal of V [[X]] and let Q ∩ V = P . If P [[X]] 6= Q and Q 6= P + (X) (i.e., X / ∈ Q), then Q is a principal ideal (and Q ⊂ P1 + (X), where P1 is the prime ideal just above P ). In a nondiscrete valuation domain or a non-SFT valuation domain, these conditions are not enough to guarantee Q to be a principal ideal (see the remark following Corollary ??). Under an additional hypothesis that Q contains a power series with unit content, we prove that Q is a principal ideal. This result will enable us to extend a part of [?, Proposition 5] to the infinite-dimensional case. We will give a characterization of prime elements of V [[X]], V an SFT valuation domain. For f ∈ V [[X]], we denote by Cf the ideal of V generated by the coefficients of f . When Cf is a unit ideal, we usually write Cf = 1. Lemma 1. (see [?, Proposition 5]) Let V be a valuation domain with the maximal ideal M and Q a prime ideal of V [[X]] such that X / ∈ Q. If Q contains an element f such that Cf = 1, then Q is a principal ideal. Proof. Let g = ∑∞ i=0 aiX i ∈ Q with Cg = 1. Let n (g) be the smallest integer such that an is a unit. Let f ∈ Q be such that n (f) is the smallest in the set {n (g) | g ∈ Q and Cg = 1}. Let a0 be the constant term of f . Note that a0 6= 0. For otherwise f = Xh, h ∈ V [[X]]. Since X / ∈ Q, h ∈ Q. However, n (h) = n (f)−1, 1991 Mathematics Subject Classification. Primary 13A15, 13C15, 13F05, 13F25.