For any ideal I of a ring R, let S=R/I. In this note, we consider the following global lifting problem: For any projective S-module Q, does there exist a projective R-module P such that Q and P/IP are isomorphic as S-...For any ideal I of a ring R, let S=R/I. In this note, we consider the following global lifting problem: For any projective S-module Q, does there exist a projective R-module P such that Q and P/IP are isomorphic as S-modules? The concept of global lifting was introduced in Wu’s paper and one of our goals there is to study the computation of K<sub>0</sub> groups. Thus in what follows, Q and P are sometimes assumed to be finitely generated in addition.展开更多
IN ref. [1] Jacobson proved the structure theorem for primitive rings with nonzero socles that R is a primitive ring withsocle S≠{0} if and only if there is a pair of dual vector spaces (M,M’) over a division ring ...IN ref. [1] Jacobson proved the structure theorem for primitive rings with nonzero socles that R is a primitive ring withsocle S≠{0} if and only if there is a pair of dual vector spaces (M,M’) over a division ring Δ such that S=F(M, M’)(?) R(?)(?)(M, M’), where (?)(M, M’)-{ω∈Ω|ωM’(?)M’, Ω is the complete ring of linear transformations of M over Δ}, F(M, M’) is the set of all linear transformations of (?)(M, M’) of finite rank. After that, some people reproved this theorem by using different methods such as those in refs. [2, 3]. In展开更多
Let k be an algebraically closed field, ∧ a finite dimensional k-algebra. According to Morita equivalence, we can always assume that ∧ is basic and connected. We denote by mod∧ the category of all finite generated ...Let k be an algebraically closed field, ∧ a finite dimensional k-algebra. According to Morita equivalence, we can always assume that ∧ is basic and connected. We denote by mod∧ the category of all finite generated left ∧-modules, and by F<sub>∧</sub> the Auslander-Reiten quiver of A. Let P<sub>1</sub>, P<sub>2</sub>,…, P<sub>n</sub> be all the indecomposable projective modules up to isomorphism. For any module M ∈ mod∧, its dimension vector is defined展开更多
Let Λ and Γ be left and right Noetherian rings and Λ U a generalized tilting module with Γ = End( Λ U ). For a non-negative integer k, if Λ U is (k - 2)-Gorenstein with the injective dimensions of Λ U and U Γ ...Let Λ and Γ be left and right Noetherian rings and Λ U a generalized tilting module with Γ = End( Λ U ). For a non-negative integer k, if Λ U is (k - 2)-Gorenstein with the injective dimensions of Λ U and U Γ being k, then the socle of the last term in a minimal injective resolution of Λ U is non-zero.展开更多
An efficient and simple method for the synthesis of chalcones was reported.In the presence of sulfurous oxychloride,the cross-aldol condensation of acetone and benzaldehyde was catalyzed by the anhydrous ethanol with ...An efficient and simple method for the synthesis of chalcones was reported.In the presence of sulfurous oxychloride,the cross-aldol condensation of acetone and benzaldehyde was catalyzed by the anhydrous ethanol with good yields(60%-95%)under mild condition.The best reaction condition was studied.The reaction mechanism was also discussed.展开更多
文摘For any ideal I of a ring R, let S=R/I. In this note, we consider the following global lifting problem: For any projective S-module Q, does there exist a projective R-module P such that Q and P/IP are isomorphic as S-modules? The concept of global lifting was introduced in Wu’s paper and one of our goals there is to study the computation of K<sub>0</sub> groups. Thus in what follows, Q and P are sometimes assumed to be finitely generated in addition.
文摘IN ref. [1] Jacobson proved the structure theorem for primitive rings with nonzero socles that R is a primitive ring withsocle S≠{0} if and only if there is a pair of dual vector spaces (M,M’) over a division ring Δ such that S=F(M, M’)(?) R(?)(?)(M, M’), where (?)(M, M’)-{ω∈Ω|ωM’(?)M’, Ω is the complete ring of linear transformations of M over Δ}, F(M, M’) is the set of all linear transformations of (?)(M, M’) of finite rank. After that, some people reproved this theorem by using different methods such as those in refs. [2, 3]. In
基金Project supported by the National Natural Science Foundation of China
文摘Let k be an algebraically closed field, ∧ a finite dimensional k-algebra. According to Morita equivalence, we can always assume that ∧ is basic and connected. We denote by mod∧ the category of all finite generated left ∧-modules, and by F<sub>∧</sub> the Auslander-Reiten quiver of A. Let P<sub>1</sub>, P<sub>2</sub>,…, P<sub>n</sub> be all the indecomposable projective modules up to isomorphism. For any module M ∈ mod∧, its dimension vector is defined
基金supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060284002)National Natural Science Foundation of China (Grant No. 10771095)Natural Science Foundation of Jiangsu Province of China (Grant No. BK2007517)
文摘Let Λ and Γ be left and right Noetherian rings and Λ U a generalized tilting module with Γ = End( Λ U ). For a non-negative integer k, if Λ U is (k - 2)-Gorenstein with the injective dimensions of Λ U and U Γ being k, then the socle of the last term in a minimal injective resolution of Λ U is non-zero.
文摘An efficient and simple method for the synthesis of chalcones was reported.In the presence of sulfurous oxychloride,the cross-aldol condensation of acetone and benzaldehyde was catalyzed by the anhydrous ethanol with good yields(60%-95%)under mild condition.The best reaction condition was studied.The reaction mechanism was also discussed.
