In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left...In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left A-module. It is proved that M is an n-Gorenstein projective left A-module if and only if A ■_(R)M and Hom_(R)(A, M) are n-Gorenstein projective left A-modules if and only if M is an n-Gorenstein projective left R-module. Furthermore, when R ■ A is a separable Frobenius extension, n-Gorenstein projective dimensions are considered.展开更多
We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extensi...We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.展开更多
Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H...Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.展开更多
Let R be a ring with an automorphismφof order two.We introduce the definition ofφ-centrosymmetric matrices.Denote by M_(n)(R)the ring of all n X n matrices over R,and by Sn(φ,R)the set of all p-centrosymmetric n...Let R be a ring with an automorphismφof order two.We introduce the definition ofφ-centrosymmetric matrices.Denote by M_(n)(R)the ring of all n X n matrices over R,and by Sn(φ,R)the set of all p-centrosymmetric n×n matrices over R for any positive integer n.We show that Sn(φ,R)■M_(n)(R)is a separable Frobenius extension.If R is commutative,then Sn(φ,R)is a cellular algebra over the invariant subring R^(φ)of R.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No. 11561061)。
文摘In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left A-module. It is proved that M is an n-Gorenstein projective left A-module if and only if A ■_(R)M and Hom_(R)(A, M) are n-Gorenstein projective left A-modules if and only if M is an n-Gorenstein projective left R-module. Furthermore, when R ■ A is a separable Frobenius extension, n-Gorenstein projective dimensions are considered.
基金supported by National Natural Science Foundation of China(Grant No.11401476)China Postdoctoral Science Foundation(Grant No.2016M591592)
文摘We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.
基金Supported by the NSF of China(1097104910971052)+1 种基金the NSF of Hebei Province(A2008000135A2009000253)
文摘Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.
基金supported by Beijing Nova Program(Z181100006218010)by Research Ability Improvement Program of BUCEA(Grant No.X22026).
文摘Let R be a ring with an automorphismφof order two.We introduce the definition ofφ-centrosymmetric matrices.Denote by M_(n)(R)the ring of all n X n matrices over R,and by Sn(φ,R)the set of all p-centrosymmetric n×n matrices over R for any positive integer n.We show that Sn(φ,R)■M_(n)(R)is a separable Frobenius extension.If R is commutative,then Sn(φ,R)is a cellular algebra over the invariant subring R^(φ)of R.
