• 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).

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.817-826
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• 2015

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (${\star}$) if for each nonzero element $a{\in}R$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (${\star}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (${\star}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (${\star}$), but the converse is false.

• 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.