• The Pure and Applied Mathematics
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• v.11 no.2
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• pp.117-125
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• 2004

Let T be an integral domain, M a nonzero maximal ideal of T, K = T/M, $\psi$: T \longrightarrow K the canonical map, D a proper subring of K, and R = $\psi^{-1}$(D) the pullback domain. Assume that for each $x \; \in T$, there is a $u \; \in T$ such that u is a unit in T and $ux \; \in R$, . In this paper, we show that R is a weakly Krull domain (resp., GWFD, AWFD, WFD) if and only if htM = 1, D is a field, and T is a weakly Krull domain (resp., GWFD, AWFD, WFD).

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.863-872
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• 2007

Let D be an integral domain with quotient field K, A a nonempty set of height-one maximal t-ideals of D, F$({\Lambda})={I{\subseteq}D|I$ is an ideal of D such that $I{\subseteq}P$ for all $P{\in}A}$, and $D_F({\Lambda})={x{\in}K|xA{\subseteq}D$ for some $A{\in}F({\Lambda})}$. In this paper, we prove that if each $P{\in}A$ is the radical of a finite type v-ideal (resp., a principal ideal), then $D_{F({\Lambda})}$ is a weakly Krull domain (resp., generalized weakly factorial domain) if and only if the intersection $D_{F({\Lambda})}={\cap}_{P{\in}A}D_P$ has finite character, if and only if $F({\Lambda})$ is a t-splitting set of ideals, if and only if $F({\Lambda})$ is v-finite.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.15-23
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• 2009

Let D be an integral domain, P be a nonzero prime ideal of D, $\{P_{\alpha}{\mid}{\alpha}{\in}{\mathcal{A}}\}$ be a nonempty set of prime ideals of D, and $\{I_{\beta}{\mid}{\beta}{\in}{\mathcal{B}}\}$ be a nonempty family of ideals of D with ${\cap}_{{\beta}{\in}{\mathcal{B}}}I_{\beta}{\neq}(0)$. Consider the following conditions: (i) If $P{\subseteq}{\cup}_{{\alpha}{\in}{\mathcal{A}}}P_{\alpha}$, then $P=P_{\alpha}$ for some ${\alpha}{\in}{\mathcal{A}}$; (ii) If ${\cap}_{{\beta}{\in}{\mathcal{B}}}I_{\beta}{\subseteq}P$, then $I_{\beta}{\subseteq}P$ for some ${\beta}{\in}{\mathcal{B}}$. In this paper, we prove that D satisfies $(i){\Leftrightarrow}D$ is a generalized weakly factorial domain of ${\dim}(D)=1{\Rightarrow}D$ satisfies $(ii){\Leftrightarrow}D$ is a weakly Krull domain of dim(D) = 1. We also study the t-operation analogs of (i) and (ii).