• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.249-254
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• 2000

We give an easy proof of the fact that every noncommutative torus $A_{\omega}$ is stably isomorphic to the noncommutative torus $C(\widehat{S\omega}){\;}\bigotimes{\;}A_p$ which hasa trivial bundle structure. It is well known that stable isomorphism of two separable $C^{*}-algebras$ is equibalent to the existence of eqivalence bimodule between the two stably isomorphic $C^{*}-algebras{\;}A_{\omega}$ and $C(\widehat{S\omega}){\;}\bigotimes{\;}A_p$.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1389-1421
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• 2018

In this paper, we introduce the notion of C-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where R and S are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of C-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.859-869
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• 1997

It is shown that for $A_{k, m}$ a k-homogeneous $C^*$-algebra over $S^{2n - 1} \times S^1$ such that no non-trivial matrix algebra can be factored out of $A_{k, m}$ and $A_{k, m} \otimes M_l(C)$ has a non-trivial bundle structure for any positive integer l, we construct an $A_{k, m^-} C(S^{2n - 1} \times S^1) \otimes M_k(C)$-equivalence bimodule to show that every k-homogeneous $C^*$-algebra over $S^{2n - 1} \times S^1)$. Moreover, we prove that the tensor product of the k-homogeneous $C^*$-algebra $A_{k, m}$ with a UHF-algebra of type $p^\infty$ has the tribial bundle structure if and only if the set of prime factors of k is a subset of the set of prime factors of pp.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1457-1482
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• 2017

Let S and R be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study C-weak flat and C-weak injective modules as a generalization of C-flat and C-injective modules ([21]) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class ${\mathcal{A}}_C$ (R) and that of the Bass class ${\mathcal{B}}_C$ (S). Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in ${\mathcal{A}}_C$ (R) and ${\mathcal{B}}_C$ (S). Finally we consider an open question which is closely relative to the main results ([11]), and discuss the relationship between the Bass class ${\mathcal{B}}_C$(S) and the class of Gorenstein injective modules.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.29-40
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• 2024

Let S and R be rings and _SC_R a semidualizing bimodule. We introduce the notion of G_C-FP_n-injective modules, which generalizes G_C-FP-injective modules and G_C-weak injective modules. The homological properties and the stability of G_C-FP_n-injective modules are investigated. When S is a left n-coherent ring, several nice properties and new Foxby equivalences relative to G_C-FP_n-injective modules are given.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.111-121
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• 2006

All d-homogeneous $C^*$-algebras $T^d$ over $\prod^{s_4}S^4{\times}\prod^{s_2}S^2{\times}\prod^{s_3}S^3{\times}\prod^{s_1}S^1$ are constructed. It is shown that $T^d$ are strongly Morita equivalent to $C(\prod^{s_4}S^4{\times}\prod^{s_2}S^2{\times}\prod^{s_3}S^3{\times}\prod^{s_1}S^1)$.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.393-402
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• 2006

We investigate relations between multiplicity of solutions and source terms in a elliptic equation. We have a concerne with a sharp result for multiplicity of a nonlinear Laplacian differential equation.