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Hwankoo Kim

Publications and source records attributed to Hwankoo Kim.

10 recordsLinked to original sources
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 Pr\"ufer $v$-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↗