• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.601-607
• /
• 2011

Let D be an integral domain, X be an indeterminate over D, and D[X] be the polynomial ring over D. We show that D is an almost weakly factorial PvMD if and only if D + XDS[X] is an integrally closed almost GCD-domain for each (saturated) multiplicative subset S of D, if and only if $D+XD_1[X]$ is an integrally closed almost GCD-domain for any t-linked overring $D_1$ of D, if and only if $D_1+XD_2[X]$ is an integrally closed almost GCD-domain for all t-linked overrings $D_1{\subseteq}D_2$ of D.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.4
• /
• pp.655-660
• /
• 1999

Let R be an integral domain with identity. In this paper we will show that if R is integrally closed or if t-dim $R{\leq}1$, then R[{$X_{\alpha}$}] satisfies the principal ideal theorem for each family {$X_{\alpha}$} of algebraically independent indeterminates if and only if R is an S-domain and it satisfies the principal ideal theorem.

• 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 ${\mid}SF(D){\mid}{\leq}2$.

• Korean Journal of Mathematics
• /
• v.17 no.2
• /
• pp.189-196
• /
• 2009

Let D be an integral domain, and let U be a group of units of D. Let $D^*=D-\{0\}$ and ${\Gamma}=D^*/U$ be the commutative cancellative semigroup under aU+bU=abU. We prove that $Cl(D)=Cl({\Gamma})$ and that D is a PvMD (resp., GCD-domain, Mori domain, Krull domain, factorial domain) if and only if ${\Gamma}$ is a PvMS(resp., GCD-semigroup, Mori semigroup, Krull semigroup, factorial semigroup). Let U=U(D) be the group of units of D. We also show that if D is integrally closed, then $D[{\Gamma}]$, the semigroup ring of ${\Gamma}$ over D, is an integrally closed domain with $Cl(D[{\Gamma}])=Cl(D){\oplus}Cl(D)$; hence D is a PvMD (resp., GCD-domain, Krull domain, factorial domain) if and only if $D[{\Gamma}]$ is.

• Kyungpook Mathematical Journal
• /
• v.61 no.1
• /
• pp.23-32
• /
• 2021

Let D be an integral domain with quotient field K, X an indeterminate over D, ∗ a star operation on D, and Cl∗ (D) be the ∗-class group of D. The ∗w-operation on D is a star operation defined by I∗w = {x ∈ K | xJ ⊆ I for a nonzero finitely generated ideal J of D with J∗ = D}. In this paper, we study two star operations {∗} and [∗] on D[X] defined by A{∗} = ∩P∈∗w-Max(D) ADP [X] and A[∗] = (∩P∈∗w-Max(D) AD[X]P[X]) ∩ AK[X]. Among other things, we show that Cl∗(D) ≅ Cl[∗](D[X]) if and only if D is integrally closed.

• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.4
• /
• pp.971-983
• /
• 2023

In this paper, we prove that a domain R is an FGV-domain if every finitely generated torsion-free R-module is strongly copure projective, and a coherent domain is an FGV-domain if and only if every finitely generated torsion-free R-module is strongly copure projective. To do this, we characterize G-Prüfer domains by G-flat modules, and we prove that a domain is G-Prüfer if and only if every submodule of a projective module is G-flat. Also, we study the D + M construction of G-Prüfer domains. It is seen that there exists a non-integrally closed G-Prüfer domain that is neither Noetherian nor divisorial.

• 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).

• Journal of the Korean Mathematical Society
• /
• v.53 no.5
• /
• pp.1057-1075
• /
• 2016

Let R be a ring in which Nil(R) is a divided prime ideal of R. Then, for a suitable property X of integral domains, we can define a ${\phi}$-X-ring if R/Nil(R) is an X-domain. This device was introduced by Badawi [8] to study rings with zero divisors with a homomorphic image a particular type of domain. We use it to introduce and study a number of concepts such as ${\phi}$-Schreier rings, ${\phi}$-quasi-Schreier rings, ${\phi}$-almost-rings, ${\phi}$-almost-quasi-Schreier rings, ${\phi}$-GCD rings, ${\phi}$-generalized GCD rings and ${\phi}$-almost GCD rings as rings R with Nil(R) a divided prime ideal of R such that R/Nil(R) is a Schreier domain, quasi-Schreier domain, almost domain, almost-quasi-Schreier domain, GCD domain, generalized GCD domain and almost GCD domain, respectively. We study some generalizations of these concepts, in light of generalizations of these concepts in the domain case, as well. Here a domain D is pre-Schreier if for all $x,y,z{\in}D{\backslash}0$, x | yz in D implies that x = rs where r | y and s | z. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in [15] where it was shown that a GCD domain is a Schreier domain.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.2
• /
• pp.525-530
• /
• 2015

Let D be an integrally closed domain with quotient field K, X be an indeterminate over D, $f=a_0+a_1X+{\cdots}+a_nX^n{\in}D[X]$ be irreducible in K[X], and $Q_f=fK[X]{\cap}D[X]$. In this paper, we show that $Q_f$ is a maximal ideal of D[X] if and only if $(\frac{a_1}{a_0},{\cdots},\frac{a_n}{a_0}){\subseteq}P$ for all nonzero prime ideals P of D; in this case, $Q_f=\frac{1}{a_0}fD[X]$. As a corollary, we have that if D is a Krull domain, then D has infinitely many height-one prime ideals if and only if each maximal ideal of D[X] has height ${\geq}2$.