Mi Hee Park

Group rings and semigroup rings over Strong Mori domains

Abstract In this paper we study the transfer of the property of being a Strong Mori domain. In particular we give the characterizations of Strong Mori domains in certain types of pullbacks. We show that if R is a Strong Mori domain which is not a field, then the polynomial ring R[{Xλ}λ∈Λ] is also a Strong Mori domain and w- dim R[{X λ } λ∈Λ ]=w - dim R . We also determine necessary and sufficient conditions in order that the group ring R[X;G] or the semigroup ring R[X;S] should be a Strong Mori domain with w-dimension ≤1.

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}.

Integral Domains Which Admit at Most Two Star Operations

In this article, we characterize domains which admit at most two star operations in the integrally closed and Noetherian cases. We also precisely count the number of star operations on an h-local Prüfer domain.

On overrings of Strong Mori domains

Abstract We show that if R is a Strong Mori domain (SM domain) and T is a w -overdomain of R such that T ⊆ R wg , then T is an SM domain; if R is an SM domain, then R S is an SM domain for any generalized multiplicative system S of R . Finally, we give an example such that an intersection with finiteness condition of SM domains is not an SM domain.

Integral Closure of Graded Integral Domains

Let Γ be a torsion-free cancellative commutative monoid and let R = ⨁α∈ΓRα be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.

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.

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.

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.

ON GRADED KRULL OVERRINGS OF A GRADED NOETHERIAN DOMAIN

Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dimR 2, then A is a graded Noetherian domain with h-dimA 2. This is a generalization of the well-known theorem that a Krull overring of a Noetherian domain with dimension 2 is also a Noetherian domain with dimension 2.