• Journal of the Korean Mathematical Society
• /
• v.54 no.1
• /
• pp.59-68
• /
• 2017

Let $F=R^{(n)}$ be a free R-module of finite rank $n{\geq}2$. In this paper, we characterize the prime submodules of F with at most n generators when R is a $Pr{\ddot{u}}fer$ domain. We also introduce the notion of prime matrix and we show that when R is a valuation domain, every finitely generated prime submodule of F with at least n generators is the row space of a prime matrix.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.625-631
• /
• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Korean Journal of Mathematics
• /
• v.18 no.4
• /
• pp.335-342
• /
• 2010

Let D be an integral domain with quotient field K,$\mathcal{I}(D)$ be the set of nonzero ideals of D, and $w$ be the star-operation on D defined by $I_w=\{x{\in}K{\mid}xJ{\subseteq}I$ for some $J{\in}\mathcal{I}(D)$ such that J is finitely generated and $J^{-1}=D\}$. The D is called a Pr$\ddot{u}$fer $v$-multiplication domain if $(II^{-1})_w=D$ for all nonzero finitely generated ideals I of D. In this paper, we show that D is a Pr$\ddot{u}$fer $v$-multiplication domain if and only if $(A{\cap}(B+C))_w=((A{\cap}B)+(A{\cap}C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$, if and only if $(A(B{\cap}C))_w=(AB{\cap}AC)_w$ for all $A,B,C{\in}\mathcal{I}(D)$, if and only if $((A+B)(A{\cap}B))_w=(AB)_w$ for all $A,B{\in}\mathcal{I}(D)$, if and only if $((A+B):C)_w=((A:C)+(B:C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$ with C finitely generated, if and only if $((a:b)+(b:a))_w=D$ for all nonzero $a,b{\in}D$, if and only if $(A:(B{\cap}C))_w=((A:B)+(A:C))_w$ for all $A,B,C{\in}\mathcal{I}(D)$ with B, C finitely generated.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.809-818
• /
• 2018

We study those integral domains in which every proper ideal can be written as an invertible ideal multiplied by a nonempty product of proper radical ideals.

• Journal of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.447-459
• /
• 2016

Let D be an integral domain, {$X_{\alpha}$} be a nonempty set of indeterminates over D, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}_1}$ be the first type power series ring over D. We show that if D is a t-SFT $Pr{\ddot{u}}fer$ v-multiplication domain, then $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_{1_{D-\{0\}}}$ is a Krull domain, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_1$ is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a Krull domain.

• Kyungpook Mathematical Journal
• /
• v.54 no.4
• /
• pp.587-593
• /
• 2014

Let D be an integral domain with quotient field K, $\bar{D}$ denote the integral closure of D in K and * be a star-operation on D. In this paper, we study the *-Nagata ring of AP*MDs. More precisely, we show that D is an AP*MD and $D[X]{\subseteq}\bar{D}[X]$ is a root extension if and only if the *-Nagata ring $D[X]_{N_*}$ is an AB-domain, if and only if $D[X]_{N_*}$ is an AP-domain. We also prove that D is a P*MD if and only if D is an integrally closed AP*MD, if and only if D is a root closed AP*MD.

• Korean Journal of Mathematics
• /
• v.20 no.2
• /
• pp.185-191
• /
• 2012

Let D be an integral domain and SF(D) be the set of star operations of finite type on D. We show that if ${\mid}SF(D){\mid}$ < ${\infty}$, then every maximal ideal of D is a $t$-ideal. We give an example of integrally closed quasi-local domains D in which the maximal ideal is divisorial (so a $t$-ideal) but ${\mid}SF(D){\mid}={\infty}$. We also study the integrally closed domains D with ${\mid}SF(D){\mid}{\leq}2$.

• Kyungpook Mathematical Journal
• /
• v.53 no.1
• /
• pp.25-35
• /
• 2013

In this paper, we introduce and study the concept of "strongly $t$-linked extensions", which is a stronger version of $t$-linked extensions of integral domains. We show that for an extension of Pr$\ddot{u}$fer $v$-multiplication domains, this concept is equivalent to that of "$w$-faithfully flat".

• Korean Journal of Mathematics
• /
• v.22 no.4
• /
• pp.591-598
• /
• 2014

Let D be an integral domain with quotient field K. In [1], the authors called D a finitely valuative domain if, for each $0{\neq}u{\in}K$, there is a saturated chain of rings $D=D_0{\varsubsetneq}D_1{\varsubsetneq}{\cdots}{\subseteq}$ $D_n=D[x]$, where x = u or $u^{-1}$. They then studied some properties of finitely valuative domains. For example, they showed that the integral closure of a finitely valuative domain is a Pr$\ddot{u}$fer domain. In this paper, we introduce the notion of finitely t-valuative domains, which is the t-operation analog of finitely valuative domains, and we then generalize some properties of finitely valuative domains.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.4
• /
• pp.1075-1081
• /
• 2020

Let R be a domain. It is proved that R is coherent when IFD(R) ⩽ 1, and R is Noetherian when IPD(R) ⩽ 1. Consequently, R is a G-Prüfer domain if and only if IFD(R) ⩽ 1, if and only if wG-gldim(R) ⩽ 1; and R is a G-Dedekind domain if and only if IPD(R) ⩽ 1.