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A NOTE ON GORENSTEIN PRÜFER DOMAINS
Hu, Kui,Wang, Fanggui,Xu, Longyu Korean Mathematical Society 2016 대한수학회보 Vol.53 No.5
In this note, we mainly discuss the Gorenstein $Pr{\ddot{u}}fer$ domains. It is shown that a domain is a Gorenstein $Pr{\ddot{u}}fer$ domain if and only if every finitely generated ideal is Gorenstein projective. It is also shown that a domain is a PID (resp., Dedekind domain, $B{\acute{e}}zout$ domain) if and only if it is a Gorenstein $Pr{\ddot{u}}fer$ UFD (resp., Krull domain, GCD domain).
FINITELY GENERATED G-PROJECTIVE MODULES OVER PVMDS
Hu, Kui,Lim, Jung Wook,Xing, Shiqi Korean Mathematical Society 2020 대한수학회보 Vol.57 No.3
Let M be a finitely generated G-projective R-module over a PVMD R. We prove that M is projective if and only if the canonical map θ : M⨂<sub>R</sub> M<sup>∗</sup> → Hom<sub>R</sub>(Hom<sub>R</sub>(M, M), R) is a surjective homomorphism. Particularly, if G-gldim(R) ⩽ ∞ and Ext<sup>i</sup><sub>R</sub>(M, M) = 0 (i ⩾ 1), then M is projective.
ON OVERRINGS OF GORENSTEIN DEDEKIND DOMAINS
Hu, Kui,Wang, Fanggui,Xu, Longyu,Zhao, Songquan Korean Mathematical Society 2013 대한수학회지 Vol.50 No.5
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.
FLAT DIMENSIONS OF INJECTIVE MODULES OVER DOMAINS
Hu, Kui,Lim, Jung Wook,Zhou, De Chuan Korean Mathematical Society 2020 대한수학회보 Vol.57 No.4
Let R be a domain. It is proved that R is coherent when IFD(R) ⩽ 1, and R is Noetherian when IPD(R) ⩽ 1. Consequently, R is a G-Prüfer domain if and only if IFD(R) ⩽ 1, if and only if wG-gldim(R) ⩽ 1; and R is a G-Dedekind domain if and only if IPD(R) ⩽ 1.
ON STRONGLY GORENSTEIN HEREDITARY RINGS
Hu, Kui,Kim, Hwankoo,Wang, Fanggui,Xu, Longyu,Zhou, Dechuan Korean Mathematical Society 2019 대한수학회보 Vol.56 No.2
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.
SOME ONE-DIMENSIONAL NOETHERIAN DOMAINS AND G-PROJECTIVE MODULES
Kui Hu,김환구,De Chuan Zhou 대한수학회 2023 대한수학회보 Vol.60 No.6
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 $(R:_KT)$ 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.
Hu Kui,Ma Xubo,Zhang Teng,Ma Xuan,Huang Zifeng,Chen Yixue 한국원자력학회 2023 Nuclear Engineering and Technology Vol.55 No.8
How to generate the precise broad group cross section is important for the fast reactor design. In this study, a fast reactor multi-group cross-section generation code MGGC2.0 are developed in-house for processing ultrafine group MATXS format library. Validation and verification are performed for MGGC2.0 code by applying the benchmarks of ICSBEP handbook, and the results of MGGC2.0 agree well with that of MCNP. The consistent PN method with critical buckling search is in good agreement that condensed with TWODANT flux and flux moment for the inner core and outer core region. For the radial blanket and reflector, two region approximation method has been applied in MGGC2.0 by using collision Probability Method neutron flux solver. The RBEC-M benchmark was used to verify the power distribution calculation, and the relative error of power distribution comparison with the reference are less than 0.8% in the fuel region and the maximum relative error is 5.58% in the reflector region. Therefore, the precise broad cross section can be generated by MGGC2.0 for fast reactor
A NOTE ON ARTINIAN LOCAL RINGS
Hu, Kui,Kim, Hwankoo,Zhou, Dechuan Korean Mathematical Society 2022 대한수학회보 Vol.59 No.5
In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, 2-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, 2-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are n-generated for some positive integer n.
Flat dimensions of injective modules over domains
Kui Hu,임정욱,De Chuan Zhou 대한수학회 2020 대한수학회보 Vol.57 No.4
Let $R$ be a domain. It is proved that $R$ is coherent when $IFD(R)\lst1$, and $R$ is Noetherian when $IPD(R)\lst1$. Consequently, $R$ is a $G$-Pr$\rm\ddot{u}$fer domain if and only if $IFD(R)\lst1$, if and only if ${\rm wG\mbox{-}gldim}(R)\lst1$; and $R$ is a $G$-Dedekind domain if and only if $IPD(R)\lst1$.
Kui Hu,Fanggui Wang,Hanlin Chen 대한수학회 2013 대한수학회보 Vol.50 No.3
A domain is called an LPI domain if every locally principal ideal is invertible. It is proved in this note that if D is a LPI domain, then D[X] is also an LPI domain. This fact gives a positive answer to an open question put forward by D. D. Anderson and M. Zafrullah.