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
This paper is a contribution to the study of the automorphism groups of 2 - （v, k, 1） designs. Let D be a 2 - （v, 23, 1） design and G a block-transitive and point-primitive group of automorphism of D. Then the soc...This paper is a contribution to the study of the automorphism groups of 2 - （v, k, 1） designs. Let D be a 2 - （v, 23, 1） design and G a block-transitive and point-primitive group of automorphism of D. Then the socle of G is not Sz（q） and 2G2（q）.展开更多
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum a（[dx, d′x]） is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then the...The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum a（[dx, d′x]） is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d, d′] is the inner derivation implemented bv a.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
Throughout this note, we assume that k is an algebraically closed field and A is a basic connected finite-dimensional algebra over k (associative, with identity). All modules over an algebra A are finitely generated l...Throughout this note, we assume that k is an algebraically closed field and A is a basic connected finite-dimensional algebra over k (associative, with identity). All modules over an algebra A are finitely generated left A-modules. We usually denote, up to isomorphism, by {P_A(a)|a∈I} the set of all indecomposable projective modules, by {E_A(a)|a∈I} the set of all indecomposable injective modules, and by {S_A(a)|a∈I} the set of all simple modules,展开更多
In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions （Ci） in modular lattices, and applications, J. Algebra Appl. 15 （2016）, http： dx.doi.org/10.1142/S0219498...In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions （Ci） in modular lattices, and applications, J. Algebra Appl. 15 （2016）, http： dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions （Cll） or （C12）. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.展开更多
Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsi...Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsis infinite) of R such that L=R for any W, whereeach is a set of countably many orthogonal idempotent elements of rank one in R. Furthermore,there exists a primitive ring R and a direct sum L=of countably many minimal right ideals Li ofR, but R has no subset B =of countably many orthogonal idempotent elements of rank one such that and B can be extended to a corresponding basis of some basis of M over A.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
In this paper, we determine the second Hochschild cohomology group for a class of self-injective algebras of tame representation type namely, which are standard one-parametric but not weakly symmetric. These were clas...In this paper, we determine the second Hochschild cohomology group for a class of self-injective algebras of tame representation type namely, which are standard one-parametric but not weakly symmetric. These were classified up to derived equivalence by Bocian, Holm and Skowroński in [1]. We connect this to the deformation of these algebras.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
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.
基金Supported by the National Natural Science Foundation of China(11271208,11471054)Supported by the Graduate Students’Scientific Research Innovation Project of Jiangsu Province Ordinary University(KYLX1213)
文摘This paper is a contribution to the study of the automorphism groups of 2 - （v, k, 1） designs. Let D be a 2 - （v, 23, 1） design and G a block-transitive and point-primitive group of automorphism of D. Then the socle of G is not Sz（q） and 2G2（q）.
文摘The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum a（[dx, d′x]） is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d, d′] is the inner derivation implemented bv a.
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Project supported by the National Natural Science Foundation of China.
文摘Throughout this note, we assume that k is an algebraically closed field and A is a basic connected finite-dimensional algebra over k (associative, with identity). All modules over an algebra A are finitely generated left A-modules. We usually denote, up to isomorphism, by {P_A(a)|a∈I} the set of all indecomposable projective modules, by {E_A(a)|a∈I} the set of all indecomposable injective modules, and by {S_A(a)|a∈I} the set of all simple modules,
文摘In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions （Ci） in modular lattices, and applications, J. Algebra Appl. 15 （2016）, http： dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions （Cll） or （C12）. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.
文摘Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsis infinite) of R such that L=R for any W, whereeach is a set of countably many orthogonal idempotent elements of rank one in R. Furthermore,there exists a primitive ring R and a direct sum L=of countably many minimal right ideals Li ofR, but R has no subset B =of countably many orthogonal idempotent elements of rank one such that and B can be extended to a corresponding basis of some basis of M over A.
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In this paper, we determine the second Hochschild cohomology group for a class of self-injective algebras of tame representation type namely, which are standard one-parametric but not weakly symmetric. These were classified up to derived equivalence by Bocian, Holm and Skowroński in [1]. We connect this to the deformation of these algebras.
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
