Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper,...Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper, we have proved that: let (?) be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite; if G ∈ (?), f(∞) ≡ f(p) . f(p)≠φ for each p ∈ π, A has an (?)-decomposition.展开更多
Let （R,m） be a Noetherian local ring. Denote by N-dimnA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dimRA = dim（R/AnnRA） for certai...Let （R,m） be a Noetherian local ring. Denote by N-dimnA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dimRA = dim（R/AnnRA） for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.展开更多
Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discus...Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.展开更多
Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules...Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.展开更多
A module satisfying the descending chain condition on cyclic submodules is called coperfect.The class of coperfect modules lies properly bet ween the class of locally artinian modules and the class of semiartinian mod...A module satisfying the descending chain condition on cyclic submodules is called coperfect.The class of coperfect modules lies properly bet ween the class of locally artinian modules and the class of semiartinian modules.Let R be a commutative ring with identity.We show that every semiartinian Ti-module is coperfect if and only if R is a T-ring.It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if m/m^(2)is a finitely genera ted R-module for every maximal ideal m of R.展开更多
Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show th...Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.展开更多
文摘Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper, we have proved that: let (?) be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite; if G ∈ (?), f(∞) ≡ f(p) . f(p)≠φ for each p ∈ π, A has an (?)-decomposition.
文摘Let （R,m） be a Noetherian local ring. Denote by N-dimnA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dimRA = dim（R/AnnRA） for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.
基金The NSF (10771152,10926094) of Chinathe NSF (09KJB110006) for Colleges and Universities in Jiangsu Provincethe Research Foundation (Q4107805) of Soochow University and the Research Foundation (Q3107852) of Pre-research Project of Soochow University
文摘Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.
文摘Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.
文摘A module satisfying the descending chain condition on cyclic submodules is called coperfect.The class of coperfect modules lies properly bet ween the class of locally artinian modules and the class of semiartinian modules.Let R be a commutative ring with identity.We show that every semiartinian Ti-module is coperfect if and only if R is a T-ring.It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if m/m^(2)is a finitely genera ted R-module for every maximal ideal m of R.
文摘Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.
