• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1057-1075
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• 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
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• v.56 no.4
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• pp.1041-1057
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• 2019

Let ${\Gamma}$ be a nonzero commutative cancellative monoid (written additively), $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}$ $R_{\alpha}$ be a ${\Gamma}$-graded integral domain with $R_{\alpha}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma}$, and $S(H)=\{f{\in}R{\mid}C(f)=R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local $Pr{\ddot{u}}fer$ domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-$Pr{\ddot{u}}fer$ domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1351-1365
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• 2020

Let S and T be w-linked extension domains of a domain R with S ⊆ T. In this paper, we define what satisfying the w_R-GD property for S ⊆ T means and what being w_R- or w-GD domains for T means. Then some sufficient conditions are given for the w_R-GD property and w_R-GD domains. For example, if T is w_R-integral over S and S is integrally closed, then the w_R-GD property holds. It is also given that S is a w_R-GD domain if and only if S ⊆ T satisfies the w_R-GD property for each w_R-linked valuation overring T of S, if and only if S ⊆ (S[u])_w satisfies the w_R-GD property for each element u in the quotient field of S, if and only if S_𝔪 is a GD domain for each maximal w_R-ideal 𝔪 of S. Then we focus on discussing the relationship among GD domains, w-GD domains, w_R-GD domains, Prüfer domains, PνMDs and Pw_RMDs, and also provide some relevant counterexamples. As an application, we give a new characterization of Pw_RMDs. We show that S is a Pw_RMD if and only if S is a w_R-GD domain and every w_R-linked overring of S that satisfies the w_R-GD property is w_R-flat over S. Furthermore, examples are provided to show these two conditions are necessary for Pw_RMDs.