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Hwankoo Kim

Publications and source records attributed to Hwankoo Kim.

18 recordsLinked to original sources
arXiv2026

Almost S-GCD domains: localization, class groups, and extensions

Let $D$ be an integral domain and let $S\subseteq D\setminus\{0\}$ be multiplicative. We study the condition that, for each nonzero $a,b\in D$, some $(a^n,b^n)_v$ is $S$-principal. A power estimate extends this binary condition to arbitrary finite families and permits synchronization of exponents. We describe the obstruction to descent from the AGCD domain $D_S$ by bounded $S$-torsion and by a finiteness condition on intersections of principal ideals. For an almost Prüfer $v$-multiplication domain, the almost $S$-GCD property, the AGCD property of $D_S$, and torsion of the $S$-class group are equivalent. In Krull domains this gives a criterion in terms of the divisor classes of height-one primes meeting $S$, and finite rational class rank reduces the multiplicative set to one element. We prove conductor transfer without a principal-contraction hypothesis, polynomial transfer in several settings, a normalization homeomorphism away from $S$, and a finite-principal-cover criterion with a Picard-group obstruction. Explicit examples distinguish localization, bounded descent, and the global and sublocal conditions.

math.AC↗
arXiv2026

Commutative rings with square-difference factor absorbing factorization

We introduce and study sdf-absorbing factorization rings (SAF-rings), in which every regular ideal is a finite product of square-difference factor absorbing ideals. We investigate their behavior under direct products, localizations, factor rings, and module idealizations, and establish connections with SP-rings, Dedekind rings, and N-rings. We also introduce strongly sdf-absorbing factorization rings (SSAF-rings), relate them to SSP-rings and almost multiplication rings, and prove that the Noetherian SSAF-rings are precisely the ZPI-rings. Applications to module idealizations and monoid rings are also presented.

math.AC↗
arXiv2026

Characterizations of t-Almost Dedekind Domains via w-Secondary Modules

Let $R$ be an integral domain and let $w$ denote the $w$-operation on $R$. A nonzero GV-torsion-free $R$-module $M$ is called $w$-secondary if, for every $r\in R$, either $M/rM$ is GV-torsion or $r^nM=0$ for some $n\in\N$. A $w$-module $M$ is said to admit a $w$-secondary representation if $M=(N_1+\cdots+N_n)_w$, where each $N_i$ is a $w$-secondary submodule of $M$. Such a representation is called minimal if the ideals $P_i=\sqrt{\Ann_R(N_i)}$ are pairwise distinct, and $M$ is called $w$-representable if it admits such a representation. If the sum can be chosen direct, then $M$ is called strongly $w$-representable. We characterize $w$-secondary modules and investigate their localization and $w$-closure properties. For a nonfield domain $R$, we prove that $R$ is $t$-almost Dedekind if and only if $\wdim R=1$ and every $w$-representable $w$-module is locally strongly $w$-representable: each of its nonzero localizations at maximal $w$-ideals is strongly representable. Global strong $w$-representability is retained when all the radical annihilators in the representation are nonzero.

math.AC↗
arXiv2026

Gorenstein Prüfer domains with non-Prüfer integral closure

There exists a one-dimensional local Gorenstein Prüfer domain whose integral closure is not Prüfer. We prove that the recently constructed coherent counterexample to the normalization problem has this stronger Gorenstein property. The argument establishes permanence of divisoriality of finitely generated ideals under ultraproducts and quotients by divided primes. It also supplies uniform matrix factorizations: in an ultraproduct of plane-curve domains, every nonzero ideal generated by $r$ elements is a cycle of a two-periodic totally acyclic complex with free terms of rank $r$. The divided prime defining the one-dimensional example is flat, so these complexes descend through the quotient and localization. A specified two-generated ideal remains noninvertible in every integral overring. Over the normalization, its associated Laurent overring has nonzero first Tor with every injective cogenerator, although both rings are Gorenstein flat over the original domain. Examples distinguish this failure from a nonnormal Gorenstein curve with Prüfer normalization and show that the periodicity conclusion is stronger than the Gorenstein Prüfer property alone.

math.AC↗
arXiv2026

An overring characterization of Gorenstein Prüfer domains

We prove that a commutative integral domain is Gorenstein Prüfer if and only if every overring is Gorenstein flat over the original domain. It is enough to test overrings that are finitely generated as algebras. The proof combines finite duality with a conductor-quotient argument. Under the overring hypothesis, the conductor quotient of a proper overring generated by two elements as a module is an IF-ring, and degree-one duality lifts first syzygies of finitely presented quotient modules to totally reflexive modules. This gives a bound of one for the Gorenstein flat dimension of every module annihilated elementwise by conductor powers. Quadratic integral elements provide such conductors at nonnormal localizations, while normal localizations give principal-ideal neighborhoods. Constructible compactness yields a finite cover, and a bounded complex of localizations combines these bounds over the original domain. Consequently every finitely generated ideal is Gorenstein flat and the domain is coherent. Examples distinguish finite-type from module-finite tests and Gorenstein flatness from ordinary flatness.

math.AC↗
arXiv2026

Annihilator Multiplication Modules: Ring Characterizations and Constructions

An $A$-module $E$ is annihilator multiplication if the annihilator of each element equals that of $IE$ for a finitely generated ideal $I$. We characterize rings for which every faithful module has this property as the commutative quasi-Frobenius rings, and rings for which every module has the property as the Artinian principal ideal rings. Both characterizations reduce to two-generated modules. Over a reduced ring with finitely many minimal primes, faithful annihilator multiplication modules are exactly the regular-torsion-free modules, and principal ideals suffice in the definition. An ascending chain condition on annihilator ideals gives finite detection of the module annihilator, yielding localization and graph rigidity results. We also prove equality of associated primes with those of the faithful quotient and establish projective tensor and trace characterizations. Over local square-zero rings, the property is equivalent to nonsingularity of a bilinear multiplication map. A projective dimension argument gives a sharp length bound, attained by explicit faithful indecomposable nonprojective modules whose endomorphism rings are computed. Every cyclic submodule of these examples embeds in the ring, although the whole module is not torsionless. Support, hereditary torsion, and amalgamation criteria connect these structural results with further module constructions.

math.AC↗
arXiv2026

Strongly multiplicative sets, idempotent localizations, and $S$-prime phenomena

A multiplicative set $S$ of a commutative ring $R$ is strongly multiplicative if every family $(s_i)_{i \in I}$ of elements of $S$ admits a common multiple in $S \cap \bigcap_{i \in I} s_iR$. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that $S$ is strongly multiplicative if and only if localization at $S$ commutes with arbitrary intersections of ideals, if and only if $R_S$ is the localization at an idempotent, and if and only if $D(S)$ is clopen in $\Spec(R)$. We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag--Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed--Malek question on the role of strong multiplicativity for chains of $S$-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for $R_S$, and connect the theory with the regular $m$-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.

math.AC↗
arXiv2026

Goldman primes and higher cyclic presentations for the small finitistic dimension

Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\sup\{depth R_{\mathfrak p}\mid \mathfrak p\in GSpec(R)\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\geq2$ and $h\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.

math.AC↗
arXiv2026

Change-of-Rings Theorems for the Small Finitistic Dimension

In this paper, we study the small finitistic dimension of a commutative ring from the viewpoint of finitistic flat homological algebra. Using the class $FPR(R)$ of modules admitting finite projective resolutions, we investigate the finitistic flat ($FT$-flat) dimension and establish several of its basic properties. We prove change-of-rings results for the $FT$-flat dimension, including quotient and polynomial extension results, as well as localization inequalities. As applications, we obtain characterizations of the small finitistic dimension in terms of $FT$-flat dimension, derive quotient and polynomial extension theorems for the small finitistic dimension, and establish local upper bounds in terms of the small finitistic dimensions of localizations.

math.AC↗
arXiv2026

Relative Faithful Exact Functors and Their Applications to Homological Modules

The notions of faithfully projective, faithfully flat, and faithfully injective modules--defined as modules for which the three classical homological functors are both faithful and exact--play fundamental roles across various areas of algebra. In this paper, we extend these notions to the setting of $w$-operation theory. By introducing the concept of $w$-faithfully exact functors, we define and investigate the notions of $w$-faithfully projective, $w$-faithfully flat, and $w$-faithfully injective modules. We establish their fundamental properties and demonstrate their effectiveness in generalizing classical results.

math.AC↗
arXiv2026

$S$-Prime and $S$-maximal ideals in trivial ring extensions of commutative rings

This paper explores the study of $S$-prime and $S$-maximal ideals in the context of trivial ring extensions $A \ltimes M$. Through counterexamples, we demonstrate that $S$-prime (resp., $S$-maximal) ideals in $A \ltimes M$ are not necessarily homogeneous, and a homogeneous $S$-prime (resp., $S$-maximal) ideal does not necessarily have the form $P \ltimes M$, where $P$ is an $S_0$-prime (resp., $S_0$-maximal) ideal of $A$. Moreover, we characterize the conditions under which an ideal $J$ (not necessarily homogeneous) in the trivial ring extension $A \ltimes M$ is $S$-prime (resp., $S$-maximal). Additionally, we demonstrate that all $S$-prime (and consequently $S$-maximal) ideals in $A \ltimes M$ are of the form $P \ltimes M$, where $P$ is an $S_0$-prime ideal of $A$, if and only if $M$ is an $S_0$-divisible $A$-module. As an application, we explore the transfer of the concepts of compactly $S$-packed rings, coprimely $S$-packed rings and $S$-$pm$-rings to the trivial ring extension. These results provide significant insights into the relation between $S$-primality and $S$-maximality in trivial ring extensions, contributing to a deeper understanding of ideal theory in this context. This work not only enriches the theoretical framework of ring structures but also advances the broader field of algebraic theory through practical examples and applications.

math.AC↗
arXiv2025

Module-Theoretic Characterizations of Prufer $v$-Multiplication Domains

We present unified $w$-theoretic characterizations of Prüfer $v$-multiplication domains (P$v$MDs). A module-theoretic perspective shows that torsion submodules are $w$-pure, and for $(w$-)$\,$finitely generated modules $M$, the canonical sequence $0\to T(M)\to M\to M/T(M)\to 0$ $w$-splits, resolving an open question of Geroldinger--Kim--Loper. In a $w$-version of Hattori-Davis theory, these conditions are equivalent to $Tor^R_2(M,N)$ being $GV$-torsion for all $R$-modules $M,N$, equivalently $w$-w.gl.dim$(R)\leq 1$, or $Tor^R_1(X,A)$ being $GV$-torsion for all $X$ and torsion-free $A$, or the Davis map $A\otimes_R B \to \mathcal T\otimes_K \mathcal S$ having $GV$-torsion kernel. From an overring viewpoint, $R$ is a P$v$MD if and only if for every $R\subseteq T\subseteq K$ and every $w$-maximal ideal $m$, the localization $R_{m}\to T_{\m}$ is a flat epimorphism, so that each overring is $w$-flat and the inclusion is $w$-epimorphic. Finally, $R$ is a P$v$MD if and only if every pure $w$-injective divisible $R$-module is injective.

math.AC↗
arXiv2022

On two versions of Cohen's theorem for modules

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated $R$-module $M$ is Noetherian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with Ann$(M)\subseteq \mathfrak{p}$, there exists a finitely generated submodule $N^\mathfrak{p}$ of $M$ such that $\mathfrak{p} M\subseteq N^\mathfrak{p}\subseteq M(\mathfrak{p})$, where $M(\mathfrak{p})=\{x\in M\mid sx\in \mathfrak{p} M $ for some $s\in R \setminus \mathfrak{p} \}$. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to those for $S$-Noetherian modules and $w$-Noetherian modules.

math.AC↗
arXiv2022

On Cohen's theorem for Artinian modules

In this paper, we prove that a finitely embedded $R$-module $M$ is Artinian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with $(0:_RM)\subseteq \mathfrak{p}$, there exists a submodule $N^\mathfrak{p}$ of $M$ such that $M/N^\mathfrak{p}$ is finitely embedded and $M[\mathfrak{p}]\subseteq N^\mathfrak{p}\subseteq (0:_M\mathfrak{p})$.

math.AC↗
arXiv2022

Uniformly $S$-Noetherian rings

Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated sub-ideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for $u$-$S$-Noetherian rings. Besides, the $u$-$S$-Noetherian properties on several ring constructions are given. The notion of $u$-$S$-injective modules is also introduced and studied. Finally, we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetherian rings.

math.AC↗
arXiv2015

Super finitely presented modules and Gorenstein projective modules

Let $R$ be a commutative ring. An $R$-module $M$ is said to be super finitely presented if there is an exact sequence of $R$-modules $\cdots\rightarrow P_n\rightarrow\cdots \rightarrow P_1\rightarrow P_0\rightarrow M\rightarrow 0$ where each $P_i$ is finitely generated projective. In this paper it is shown that if $R$ has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if $R$ has the property (C) that every super finitely presented module has finite projective dimension, then $R$ is $K_0$-regular, i.e., $K_0(R[x_1,\cdots,x_n])\cong K_0(R)$ for all $n\geq 1$.

math.AC↗