Byung Gyun Kang

A localization of a power series ring over a valuation domain

Abstract Let V be a valuation domain. It is known that V〚X 1 ,…,X n 〛 V−(0) is an n -dimensional Noetherian UFD if V has a height 1 prime ideal P and P ≠ P 2 . We show that V〚X 1 ,…,X n 〛 V−(0) is an n -dimensional Noetherian regular local ring if V does not have a height 1 prime ideal. If V has a height 1 prime ideal that is idempotent, then dim V〚X 1 ,…,X n 〛 V−(0) =∞ and t- dim V〚X 1 ,…,X n 〛 V-(0) =∞ . In the process of obtaining the above result, we introduce a product of infinitely many power series.

Anti-archimedean rings and power series rings

We define an integral domain D to be anti-Archimedean if . For example, a valuation domain or SFT Prufer domain is anti-Archimedean if and only if it has no height-one prime ideals. A number of constructions and stability results for anti-Archimedean domains are given. We show that D is anti-Archimedean is quasilocal and in this case is actually an n-dimensional regular local ring. We also show that if D is an SFT Prufer domain, then is a Krull domain for any set of indeterminates {X α}.

Lifting up an infinite chain of prime ideals to a valuation ring

We prove that for an arbitrary chain {Pα} of prime ideals in an integral domain, there exists a valuation domain which has a chain of prime ideals {Qα} lying over {Pα}. It is well-known that for a domain D and a chain of prime ideals P0 ( P1 ( · · · ( Pl, there exists a valuation domain V containing D and a chain of prime ideals Q0 ( Q1 ( · · · ( Ql lying over P0 ( P1 ( · · · ( Pl [3, Corollary 19.7]. This result is crucial in proving various results. The problem about a chain of prime ideals with an arbitrary length was posed by D. D. Anderson [1, #7, p. 364]. In this paper, we answer this question affirmatively and prove that the above theorem [3, Corollary 19.7] is true for a chain of prime ideals of an arbitrary ordinal type. Throughout this paper, let D be an integral domain with quotient field K. Theorem. Let D be an integral domain and let {Pα} be a chain of prime ideals in D. Then there exists a valuation overring V of D on K with a chain of prime ideals {Qα} such that Qα ∩D = Pα. Corollary. Let {Pα} be a chain of prime ideals in an integral domain D. Then there exists, in the integral closure of D, a chain of prime ideals {Qα} lying over {Pα}. In order to prove the theorem, we need some lemmas. Lemma 1. Let {Pα}α∈Λ be a chain of prime ideals of D. Let Sα = (D − Pα)−1 Pα and S = ⋃ α Sα. Then (1) S is closed under multiplication and S is closed with respect to the multiplication by elements of D; (2) Sα is closed under addition. Proof. (1) Suppose that a, b ∈ S and c ∈ D. Let a = g f , b = k h , where a ∈ Sα, b ∈ Sβ. We may assume that Pα ( Pβ by symmetry. Then fh 6∈ Pα since f 6∈ Pα and h 6∈ Pβ . Since g ∈ Pα, gk ∈ Pα. Hence ab ∈ Sα ⊆ S. Since cg ∈ Pα, ca ∈ Sα ⊆ S. Hence S is closed under multiplication and with respect to the multiplication by elements of D. (2) This follows because Sα is the maximal ideal of the ring (D − Pα)−1D. Received by the editors May 16, 1996 and, in revised form, July 28, 1996. 1991 Mathematics Subject Classification. Primary 13A18; Secondary 13B02. This research was supported by the research grant BSRI-95-1431. c ©1998 American Mathematical Society

Characterizations of Krull rings with zero divisors

Abstract We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.

A Note on t-SFT-rings

We define a nonzero ideal A of an integral domain R to be a t-SFT-ideal if there exist a finitely generated ideal B ⊆ A and a positive integer k such that a k ∈ B v for each a ∈ A t , and a domain R to be a t-SFT-ring if each nonzero ideal of R is a t-SFT-ideal. This article presents a number of basic properties and stability results for t-SFT-rings. We show that an integral domain R is a Krull domain if and only if R is a completely integrally closed t-SFT-ring; for an integrally closed domain R, R is a t-SFT-ring if and only if R[X] is a t-SFT-ring; if R is a t-SFT-domain, then t − dim R[[X]] ≥ t − dim R. We also give an example of a t-SFT Prüfer v-multiplication domain R such that t − dim R[[X]] > t − dim R.

Lifting up a tree of prime ideals to a going-up extension

We prove that if R⊆D is an extension of commutative rings with identity and the going-up property (for example, an integral extension), then any tree T of prime ideals of R can be embedded in Spec(D), i.e., T can be covered by some isomorphic tree T′ of prime ideals of D. In particular, the prime spectrum of a Prufer domain can always be embedded in the prime spectrum of its integral extension. The most interesting case is when the integral extension is also a Prufer domain. In this case, we obtain two Prufer domains such that Spec(R)↪Spec(D). We also prove that for an integral domain R, there exists a Bezout domain D such that any tree T⊆Spec(R) can be embedded in Spec(D). We give a sufficient condition for the question: given an extension A⊆B of commutative rings and a tree T⊆Spec(B), what are necessary and sufficient conditions that Tc={Q∩A|Q∈T} be a tree in Spec(A)? We also prove that if R is an integral domain with the following property: for a given tree T in Spec(R), there exists a Prufer overring P(R) of R with the tree T′ such that (T′)c=T and T≅T′, then an integral and mated extension of R has the same property.

Delsarte's Linear Programming Bound for Constant-Weight Codes

We give an alternative proof of Delsartes linear programming bound for binary codes and its improvements. Applying the technique which is used in the proof to binary constant-weight codes, we obtain new upper bounds on sizes of binary constant-weight codes.

GCD DOMAINS AND POWER SERIES RINGS

ABSTRACT We show that if is a valuation domain with a GCD domain, then must be of rank one with value group either or . But we give an example of a rank-one valuation domain with value group for which is not a GCD domain.

Noetherian Property of Subrings of Power Series Rings

Let R be a commutative ring with unit. We study subrings R[X; Y, λ] of R[X][[Y]] = R[X 1,…, X n ][[Y 1,…, Y m ]], where λ is a nonnegative real-valued increasing function. These rings R[X; Y, λ] are obtained from elements of R[X][[Y]] by bounding their total X-degree above by λ on their Y-degree. Such rings naturally arise from studying p-adic analytic variation of zeta functions over finite fields. Under certain conditions, Wan and Davis showed that if R is Noetherian, then so is R[X; Y, λ]. In this article, we give a necessary and sufficient condition for R[X; Y, λ] to be Noetherian when Y has more than one variable and λ grows at least as fast as linear. It turns out that the ring R[X; Y, λ] is not Noetherian for a quite large class of functions λ including functions that were asked about by Wan.

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A↪B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N v = {f ∈ D[X] | c(f)−1 = D}. In this article, we study when the Nagata ring D[X] N (more generally, D[X] N v ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] N v is a GPVD if and only if D is a t-GPVD and D[X] N v has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] N v is a GPVD. As a corollary, we have that D[X] N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] N v is a GPVD.