arXiv subjects

Gyu Whan Chang

Publications and source records attributed to Gyu Whan Chang.

14 recordsLinked to original sources

arXiv2026

Approximating DVRs by elements of bounded ramification

Let $V$ be a DVR with quotient field $K$ and perfect residue field, $v$ be the valuation on $K$ associated with $V$, $\widehat K$ be the completion of $K$, and $\mathbb{K}$ be the completion of an algebraic closure $\overline{\widehat{K}}$ of $\widehat K$. We show that a DVR of the rational function field $K(X)$ which is a residually algebraic extension of $V$ is necessarily of the form $V_α=\{ϕ\in K(X)\mid v(ϕ(α))\geq0\}$, for an element $α$ of $\mathbb{K}$ transcendental over $K$, and that $α$ is algebraic over $\widehat K$ if and only if the residue field extension is finite. Not every such $V_α$ is a DVR, however, and we characterize the $α\in\mathbb{K}$ for which $V_α$ is a DVR: they are the elements which can be approximated by algebraic elements in $\overline{\widehat{K}}$ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of $V$ to $K(X)$ which are DVRs and residually algebraic over $V$, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of $α\in\mathbb K$ for which $V_α$ is a DVR is a subfield of $\mathbb K$, which sits properly between $\overline{\widehat{K}}$ and $\mathbb K$, and corresponds to those elements $α$ for which the value group of $\widehat K(α)$ is discrete.

math.AC↗

arXiv2026

Factorization in almost Dedekind domain

Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\frac{1}{p^n}}, X^{-\frac{1}{p^n}}]$ for each integer $n \geq 0$ and $D = \bigcup\limits_{n\in\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{é}zout domain but not a Dedekind domain, and $D$ is an almost Dedekind domain if and only if char$(F) \neq p$. In this paper, we study the element-wise factorization properties of $D$. For example, we determine when an irreducible element of $D_n$ is an irreducible element of $D$, in terms of $n$ and $p$. In particular, we show that if $F$ is algebraically closed or a finite field of char$(F)=p$, then $D$ has no irreducible element. We also show that if $F$ is a finite field of odd characteristic, then an irreducible element $f(X)$ of $D_0$ is irreducible in $D$ if and only if it is a factor of a cyclotomic polynomial $Φ_n(X)$ for some integer $n \geq 1$ which satisfies a certain equation in terms of $|F|$ and deg$(f(X))$. Finally, we introduce the notion of infinite product and we then show that if $F= \mathbb{Q}$ and $p=2$, every nonzero nonunit of $D$ can be written as a product of countably many prime elements of $D$ and every proper nonzero principal ideal of $D$ can be uniquely written as a countable intersection of principal primary ideals.

math.AC↗

arXiv2026

Geometric configuration of integrally closed Noetherian domains

In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over $\mathbb{Z}_{(p)}$ for some prime $p \in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in $\mathbb{C}_p$, the completion of the algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of $\mathbb{C}_p$ to its valuation domain $\mathbb{O}_p$, as $p\in\mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.

math.AC↗

arXiv2025

Valuation Ideal Factorization Domains

An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $π$-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the $*$-operation analogs of VIFDs. Among them, we show that if $D$ is treed (resp., $*$-treed), then $D$ is a VIFD (resp., $*$-VIFD) if and only if $D$ is an ${\rm h}$-local Prüfer domain (resp., a $*$-${\rm h}$-local P$*$MD) if and only if every nonzero prime ideal of $D$ contains an invertible (resp., a $*$-invertible) valuation ideal. We also study integral domains $D$ such that for each nonzero nonunit $a\in D$, there is a positive integer $n$ such that $a^n$ can be written as a finite product of valuation elements.

math.AC↗

arXiv2023

On Dedekind domains whose class groups are direct sums of cyclic groups

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$, there exists a submonoid $S_i \subseteq D^{\bullet}$ with $\Pic (D_{S_i}) \cong G_i$. \item Each class of $\Pic (D)$ and of all $\Pic (D_{S_i})$ contains infinitely many prime ideals. \end{enumerate} Furthermore, we study orders as well as sets of lengths in the Dedekind domain $D$ and in all its localizations $D_{S_i}$.

math.AC↗

arXiv2023

Prime Factorization of ideals in commutative rings, with a focus on Krull rings

Let $R$ be a commutative ring with identity. The structure theorem says that $R$ is a PIR (resp., UFR, general ZPI-ring, $π$-ring) if and only if $R$ is a finite direct product of PIDs (resp., UFDs, Dedekind domains, $π$-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations $v$ or $t$ as follows: An integral domain $R$ is a Krull domain if and only if every nonzero proper principal ideal of $R$ can be written as a finite $v$- or $t$-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation $u$ on $R$, so that $R$ is a general Krull ring if and only if every proper principal ideal of $R$ can be written as a finite $u$-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

math.AC↗

arXiv2022

Semigroup rings as weakly Krull domains

Let $D$ be an integral domain and $Γ$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[Γ]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $Γ$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.

math.AC↗

arXiv2020

Unique factorization property of non-unique factorization domains II

Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring over $D$, is a VFD and (iii) a VFD $D$ is a weakly factorial GCD-domain if and only if $D$ is archimedean. We also study a unique factorization property of VFDs.

math.AC↗

arXiv2017

Graded integral domains which are UMT-domains

Let $Γ$ be a torsionless commutative cancellative monoid, $R =\bigoplus_{α\in Γ}R_α$ be a $Γ$-graded integral domain, and $H$ be the set of nonzero homogeneous elements of $R$. In this paper, we show that if $Q$ is a maximal $t$-ideal of $R$ with $Q \cap H = \emptyset$, then $R_Q$ is a valuation domain. We then use this result to give simple proofs of the facts that (i) $R$ is a UMT-domain if and only if $R_Q$ is a quasi-Prüfer domain for each homogeneous maximal $t$-ideal $Q$ of $R$ and (ii) $R$ is a P$v$MD if and only if every nonzero finitely generated homogeneous ideal of $R$ is $t$-invertible, if and only if $R_Q$ is a valuation domain for all homogeneous maximal $t$-ideals $Q$ of $R$. Let $D[Γ]$ be the monoid domain of $Γ$ over an integral domain $D$. We also show that $D[Γ]$ is a UMT-domain if and only if $D$ is a UMT-domain and the integral closure of $Γ_S$ is a valuation monoid for all maximal $t$-ideals $S$ of $Γ$. Hence, $D[Γ]$ is a P$v$MD if and only if $D$ is a P$v$MD and $Γ$ is a P$v$MS.

math.AC↗

arXiv2013

Factorization in the self-idealization of a PID

Let $D$ be a principal ideal domain and $R(D) = \{\begin{pmatrix} a & b 0 & a \end{pmatrix} \mid a, b \in D\}$ be its self-idealization. It is known that $R(D)$ is a commutative noetherian ring with identity, and hence $R(D)$ is atomic (i.e., every nonzero nonunit can be written as a finite product of irreducible elements). In this paper, we completely characterize the irreducible elements of $R(D)$. We then use this result to show how to factorize each nonzero nonunit of $R(D)$ into irreducible elements. We show that every irreducible element of $R(D)$ is a primary element, and we determine the system of sets of lengths of $R(D)$.

math.AC↗

arXiv2013

Polynomial extensions of semistar operations

We provide a complete solution to the problem of extending arbitrary semistar operations of an integral domain $D$ to semistar operations of the polynomial ring $D[X]$. As an application, we show that one can reobtain the main results of some previous papers concerning the problem in the special cases of stable semistar operations of finite type or semistar operations defined by families of overrings. Finally, we investigate the behavior of the polynomial extensions of the most important and classical operations such as $d_D$, $v_D$, $t_D$, $w_D$ and $b_D$ operations.

math.AC↗

arXiv2010

An overring-theoretic approach to polynomial extensions of star and semistar operations

Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$ (respectively, $E^\star [X]= (E[X])^{\ast}$) for all nonzero $D$-submodules $E$ of $K$. In this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on $D[X]$ that are "canonical" extensions (or, "canonical" strict extensions) of semistar operations on $D$.

math.AC↗

arXiv2008

Uppers to zero in polynomial rings and Prüfer-like domains

Let $D$ be an integral domain and $X$ an indeterminate over $D$. It is well known that (a) $D$ is quasi-Prüfer (i.e, its integral closure is a Prüfer domain) if and only if each upper to zero $Q$ in $D[X] $ contains a polynomial $g \in D[X]$ with content $\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \in D[X]$ with $\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this paper, given a semistar operation $\star$ in the sense of Okabe-Matsuda, we introduce the $\star$-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UM$t$-domains and the Prüfer $v$-multiplication domains.

math.AC↗

arXiv2007

Uppers to zero and semistar operations in polynomial rings

Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Prüfer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Prüfer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Prüfer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.

math.AC↗