Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=...Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=cid(M R) and if S M is a left S-module then cfd(S M)=cfd(R M).Moreover,cid-D(S)=cid-D(R) and cfd-D(S)=cfdD(R).展开更多
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
In Section 1 of this paper,we investigate the finitely presented dimension of an essential extension for a module,and obtain results concerning an essential extension of a torsion-free module. We partially answer the ...In Section 1 of this paper,we investigate the finitely presented dimension of an essential extension for a module,and obtain results concerning an essential extension of a torsion-free module. We partially answer the question:When is an essential extension of a finitely presented module(an almost finitely presented module)also finitely presented(almost finitely presented)?In Section 2,we study the C-excellent extensions and the finitely presented dimensions.We obtain some results on the homological dimensions of matrix rings and group rings.展开更多
In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, poly...In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.展开更多
Introduction Extensive Reading, as the name indicates, is a guide for students into the vast sea of knowledge. Reading demonstrates to students that there is no end in learning knowledge. However, reading and language...Introduction Extensive Reading, as the name indicates, is a guide for students into the vast sea of knowledge. Reading demonstrates to students that there is no end in learning knowledge. However, reading and language learning based on textbooks is far from enough because the knowledge from them is too narrow, the vocabulary is too limited and the contents are not current enough in these times of information exchange. Throughout my teaching experiments, I have found that newspaper clippings can become a very good way to bring more benefits to students’ learning than I expected. It can broaden the students’ horizons,展开更多
Open-source large language models(LLMs)research has made significant progress,but most studies predominantly focus on general-purpose English data,which poses challenges for LLM research in Chinese education.To addres...Open-source large language models(LLMs)research has made significant progress,but most studies predominantly focus on general-purpose English data,which poses challenges for LLM research in Chinese education.To address this,this research first reviewed and synthesized the core technologies of representative open-source LLMs,and designed an advanced 1.5Bparameter LLM tailored for the Chinese education field.Chinese education large language model(CELLM)is trained from scratch,involving two stages,namely,pretraining and instruction fine-tuning.In the pre-training phase,an open-source dataset is utilized for the Chinese education domain.During the instruction fine-tuning stage,the Chinese instruction dataset is developed and open-sourced,comprising over 258,000 data entries.Finally,the results and analysis of CELLM across multiple evaluation datasets are presented,which provides a reference baseline performance for future research.All of the models,data,and codes are opensource to foster community research on LLMs in the Chinese education domain.展开更多
This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to t...This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.展开更多
We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian ...We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause＇s work [Math. Ann., 2012, 353： 765-781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.展开更多
文摘Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=cid(M R) and if S M is a left S-module then cfd(S M)=cfd(R M).Moreover,cid-D(S)=cid-D(R) and cfd-D(S)=cfdD(R).
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.
基金Supported by the Natural Science Foundation of China
文摘In Section 1 of this paper,we investigate the finitely presented dimension of an essential extension for a module,and obtain results concerning an essential extension of a torsion-free module. We partially answer the question:When is an essential extension of a finitely presented module(an almost finitely presented module)also finitely presented(almost finitely presented)?In Section 2,we study the C-excellent extensions and the finitely presented dimensions.We obtain some results on the homological dimensions of matrix rings and group rings.
基金Supported by the NNSF of China(10901129)Supported by the SRFDP(20096203120001)
文摘In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.
文摘Introduction Extensive Reading, as the name indicates, is a guide for students into the vast sea of knowledge. Reading demonstrates to students that there is no end in learning knowledge. However, reading and language learning based on textbooks is far from enough because the knowledge from them is too narrow, the vocabulary is too limited and the contents are not current enough in these times of information exchange. Throughout my teaching experiments, I have found that newspaper clippings can become a very good way to bring more benefits to students’ learning than I expected. It can broaden the students’ horizons,
文摘Open-source large language models(LLMs)research has made significant progress,but most studies predominantly focus on general-purpose English data,which poses challenges for LLM research in Chinese education.To address this,this research first reviewed and synthesized the core technologies of representative open-source LLMs,and designed an advanced 1.5Bparameter LLM tailored for the Chinese education field.Chinese education large language model(CELLM)is trained from scratch,involving two stages,namely,pretraining and instruction fine-tuning.In the pre-training phase,an open-source dataset is utilized for the Chinese education domain.During the instruction fine-tuning stage,the Chinese instruction dataset is developed and open-sourced,comprising over 258,000 data entries.Finally,the results and analysis of CELLM across multiple evaluation datasets are presented,which provides a reference baseline performance for future research.All of the models,data,and codes are opensource to foster community research on LLMs in the Chinese education domain.
基金Supported by the National Natural Science Foundation of China (Grant No.10961021)
文摘This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.
基金Acknowledgements The author would like to thank his supervisor Professor Zhaoyong Huang, for the valuable help, suggestions, guidance, and encouragement during his studies and preparation of this paper. He also thanks the referees for their careful reading and for pointing out related references. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11571164）.
文摘We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause＇s work [Math. Ann., 2012, 353： 765-781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.
