• Journal of the Korean Mathematical Society
• /
• v.45 no.5
• /
• pp.1221-1241
• /
• 2008

Most of domains people have studied are convex bounded projective (or affine) domains. Edith $Soci{\acute{e}}$-$M{\acute{e}}thou$ [15] characterized ellipsoid in ${\mathbb{R}}^n$ by studying projective automorphism of convex body. In this paper, we showed convex and bounded projective domains can be identified from local data of their boundary points using scaling technique developed by several mathematicians. It can be found that how the scaling technique combined with properties of projective transformations is used to do that for a projective domain given local data around singular boundary point. Furthermore, we identify even unbounded or non-convex projective domains from its local data about a boundary point.

• 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.

• Journal of the Korean Mathematical Society
• /
• v.50 no.5
• /
• pp.991-1008
• /
• 2013

In this paper, we mainly discuss Gorenstein Dedekind do-mains (G-Dedekind domains for short) and their overrings. Let R be a one-dimensional Noetherian domain with quotient field K and integral closure T. Then it is proved that R is a G-Dedekind domain if and only if for any prime ideal P of R which contains ($R\;:_K\;T$), P is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a 2-DVR. As an application, we prove that a Noetherian domain R is a Warfield domain if and only if for any maximal ideal M of R, $R_M$ is a 2-DVR.

• Journal of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.427-440
• /
• 2017

In this paper, in terms of the notions of strongly copure projective modules and the copure projective dimension cpD(R) of a ring R were defined in [12], we show that a domain R has $cpD(R){\leq}1$ if and only if R is a Gorenstein Dedekind domain.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.649-657
• /
• 2018

Let R be a commutative ring. In this paper, the w-projective Basis Lemma for w-projective modules is given. Then it is shown that for a domain, nonzero w-projective ideals and nonzero w-invertible ideals coincide. As an application, it is proved that R is a Krull domain if and only if every submodule of finitely generated projective modules is w-projective.

• Journal of the Korean Mathematical Society
• /
• v.59 no.4
• /
• pp.821-841
• /
• 2022

In this paper, we introduce and study regular rings relative to the hereditary torsion theory w (a special case of a well-centered torsion theory over a commutative ring), called w-regular rings. We focus mainly on the w-regularity for w-coherent rings and w-Noetherian rings. In particular, it is shown that the w-coherent w-regular domains are exactly the Prüfer v-multiplication domains and that an integral domain is w-Noetherian and w-regular if and only if it is a Krull domain. We also prove the w-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, we show that every w-Noetherian w-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak w-projective dimension of a w-Noetherian ring is 0, 1, or ∞.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.1
• /
• pp.131-135
• /
• 2011

It is shown that generalized GCD domains satisfy a certain property of injectivity and conversely this property characterizes generalized GCD domains.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1879-1886
• /
• 2016

In this paper, we investigate domains having a flat piece in the boundary and find a sufficient condition for such a domain to be a cone.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.373-382
• /
• 2019

In this note, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of SG-projective modules and the class of G-projective modules coincide if and only if the class of SG-projective modules is closed under extension. From this we get that a ring is an SG-hereditary ring if and only if every ideal is G-projective and the class of SG-projective modules is closed under extension. We also give some examples of domains whose ideals are SG-projective.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1453-1461
• /
• 2023

Let R be a one-dimensional Noetherian domain with quotient field K and T be the integral closure of R in K. In this note we prove that if the conductor ideal (R :_K T) is a nonzero prime ideal, then every finitely generated reflexive (and hence finitely generated G-projective) R-module is isomorphic to a direct sum of some ideals.