• Journal of the Korean Mathematical Society
• /
• v.38 no.5
• /
• pp.915-935
• /
• 2001

Let V be any valuation domain and let E be a subset of the quotient field K of V. We study the ring of integer-valued polynomials on E, that is, Int(E, V)={f$\in$K[X]｜f(E)⊆V}. We show that, if E is precompact, then Int(E, V) has many properties similar to those of the classical ring Int(Z).In particular, Int(E, V) is dense in the ring of continuous functions C(E, V); each finitely generated ideal of Int(E, V) may be generated by two elements; and finally, Int(E, V) is a Prufer domain.

• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.69-77
• /
• 2023

In this article we relate the six Prüfer conditions with the EM conditions. We use the EM-conditions to prove some cases of equivalence of the six Prüfer conditions. We also use the Prüfer conditions to answer some open problems concerning EM-rings.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.259-268
• /
• 1998

We give several characterizations of a TV-PVMD and we show that the localization R[X;S]$_{N_{\upsilon}}$ of a semigroup ring R[X;S] is a TV-PVMD if and only if R is a TV-PVMD where $N_{\upsilon}\;=\;\{f\;{\in}\;R[X]{\mid}(A_f)_{\upsilon} = R\}$ and S is a torsion free cancellative semigroup with zero.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.1
• /
• pp.213-226
• /
• 2022

Let R be a ring and M an R-module. Then M is said to be regular w-flat provided that the natural homomorphism I ⊗_R M → R ⊗_R M is a w-monomorphism for any regular ideal I. We distinguish regular w-flat modules from regular flat modules and w-flat modules by idealization constructions. Then we give some characterizations of total quotient rings and Prüfer v-multiplication rings (PvMRs for short) utilizing the homological properties of regular w-flat modules.

• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.11-19
• /
• 2023

Let 𝓗₀ be the set of rings R such that Nil(R) = Z(R) is a divided prime ideal of R. The concept of maximal non φ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in 𝓗₀. This generalization was introduced in [20] where the authors proved that if R ∈ 𝓗₀ is an integrally closed ring with finite Krull dimension, then R is a maximal non φ-chained subring of T(R) if and only if R is not local and |[R, T(R)]| = dim(R) + 3. This motivates us to investigate the other natural numbers n for which R is a maximal non φ-chained subring of some overring S. The existence of such an overring S of R is shown for 3 ≤ n ≤ 6, and no such overring exists for n = 7.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.5
• /
• pp.1013-1018
• /
• 2009

Let * be an e.a.b. star operation on an integrally closed domain D, and let $K\gamma$(D, *) be the Kronecker function ring of D. We show that if D is a P*MD, then the mapping $D_{\alpha}{\mapsto}K{\gamma}(D_{\alpha},\;{\upsilon})$ is a bijection from the set {$D_{\alpha}$} of *-linked overrings of D into the set of overrings of $K{\gamma}(D,\;{\upsilon})$. This is a generalization of [5, Proposition 32.19] that if D is a Pr$\ddot{u}$fer domain, then the mapping $D_{\alpha}{\mapsto}K_{\gamma}(D_{\alpha},\;b)$ is a one-to-one mapping from the set {$D_{\alpha}$} of overrings of D onto the set of overrings of $K_{\gamma}$(D, b).