• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.935-946
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• 2015

The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be ${\phi}$-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map ${\phi}$ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a ${\phi}$-ring R is said to be a ${\phi}$-pseudo-strongly prime ideal if, whenever $x,y{\in}R_{Nil(R)}$ and $(xy){\phi}(P){\subseteq}{\phi}(P)$, then there exists an integer $m{\geqslant}1$ such that either $x^m{\in}{\phi}(R)$ or $y^m{\phi}(P){\subseteq}{\phi}(P)$. If each prime ideal of R is a ${\phi}$-pseudo strongly prime ideal, then we say that R is a ${\phi}$-pseudo-almost valuation ring (${\phi}$-PAVR). Among the properties of ${\phi}$-PAVRs, we show that a quasilocal ${\phi}$-ring R with regular maximal ideal M is a ${\phi}$-PAVR if and only if V = (M : M) is a ${\phi}$-almost chained ring with maximal ideal $\sqrt{MV}$. We also investigate the overrings of a ${\phi}$-PAVR.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.21-28
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• 2001

In this short paper, we generalize some theorems about pseudo valuation domain to ring and give characterizations of psedo valuation ring.

• Journal of the Korean Mathematical Society
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• v.61 no.5
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• pp.953-973
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• 2024

Let R = ⊕_α∈Γ R_α be a commutative ring graded by an arbitrary torsionless monoid Γ. A homogeneous prime ideal P of R is said to be strongly homogeneous prime if aP and bR are comparable for any homogeneous elements a, b of R. We will say that R is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of R is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary Γ-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of Γ-graded rings that satisfy the above mentioned property.

• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.57-67
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• 2022

In the present study, we introduce a valuation of ternary semiring on an ordered abelian group. Motivated by the construction of valuation rings, we study some properties of ideals in ternary semiring arising in connection with the valuation map. We also explore ternary valuation semirings for a noncommuative ternary division semiring. We further consider the notion of convexity in a ternary semiring and how it is reflected in the valuation map.

• Korean Journal of Mathematics
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• v.15 no.2
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• pp.115-119
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• 2007

Let R be a Krull ring with the unique regular maximal ideal M. We show that R has a regular prime element and reg-$dimR=1{\Leftrightarrow}R$ is a factorial ring and reg-$dim(R)=1{\Rightarrow}M$ is invertible ${\Leftrightarrow}R{\varsubsetneq}[R:M]{\Leftrightarrow}M$ is divisorial ${\Leftrightarrow}$ reg-$htM=1{\Rightarrow}R$ is a rank one discrete valuation ring. We also show that if M is generated by regular elements, then R is a rank one discrete valuation ring ${\Rightarrow}$ R is a factorial ring and reg-dim(R)=1.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.71-77
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• 2024

Let A be a commutative integral domain with identity element and S a multiplicatively closed subset of A. In this paper, we introduce the concept of S-valuation domains as follows. The ring A is said to be an S-valuation domain if for every two ideals I and J of A, there exists s ∈ S such that either sI ⊆ J or sJ ⊆ I. We investigate some basic properties of S-valuation domains. Many examples and counterexamples are provided.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1095-1106
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• 2020

The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let 𝓗 = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. Let R ∈ 𝓗 be a ring with total quotient ring T(R) and define 𝜙 : T(R) → R_Nil(R) by ${\phi}({\frac{a}{b}})={\frac{a}{b}}$ for any a ∈ R and any regular element b of R. Then 𝜙 is a ring homomorphism from T(R) into R_Nil(R) and 𝜙 restricted to R is also a ring homomorphism from R into R_Nil(R) given by ${\phi}(x)={\frac{x}{1}}$ for every x ∈ R. We say that R is a 𝜙-pseudo-Krull ring if 𝜙(R) = ∩ R_i, where each R_i is a nonnil-Noetherian 𝜙-pseudo valuation overring of 𝜙(R) and for every non-nilpotent element x ∈ R, 𝜙(x) is a unit in all but finitely many R_i. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.7-11
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• 1994

Classically, valuation theory is closely related to the theory of divisors and conversely. If D is a Dedekined ring and K is its quotient field, then we can clearly construct the theory of divisors on D (or K), and then we can induce all the valuations on K ([3]). In particular, if K is a number field and A is the ring of algebraic integers, then since Z is Dedekind, A is a Dedekind rign and K is the field of fractions of A.(omitted)

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

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.1-10
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• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.