Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In thi...Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In this paper,we prove that Hom S(_(B)B_(S),_(R)C_(S))is a semidualizing(A,B)-bimodule under some suitable conditions,and so Hom S(_(B)B_(S),_(R)C_(S))induces an equivalence between the Auslander class AHomS (_(B)B_(S),_(R)C_(S))(B). and the Bass class BHomS (BBS,RCS)(A) Furthermore,under a suitable condition on_(R)C_(S),we develop a generalized Morita theory for Auslander categories.展开更多
Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-inject...Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v（R） which leads to the fact that V-Gorenstein injective modules admit exact right Iv （R）-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv （R） （I, N） = 0 for all modules I with finite Iv （R）-injective dimension.展开更多
基金Supported by the Natural Science Foundation of Anhui Province(Grant No.2008085QA03)。
文摘Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In this paper,we prove that Hom S(_(B)B_(S),_(R)C_(S))is a semidualizing(A,B)-bimodule under some suitable conditions,and so Hom S(_(B)B_(S),_(R)C_(S))induces an equivalence between the Auslander class AHomS (_(B)B_(S),_(R)C_(S))(B). and the Bass class BHomS (BBS,RCS)(A) Furthermore,under a suitable condition on_(R)C_(S),we develop a generalized Morita theory for Auslander categories.
文摘Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v（R） which leads to the fact that V-Gorenstein injective modules admit exact right Iv （R）-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv （R） （I, N） = 0 for all modules I with finite Iv （R）-injective dimension.
