Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), ...Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), f(p)≠ for each p∈π. The following conclutions are obtained: (1) if there exists a maximal submodule B of A such that A/B is F-central in G and B has no nonzero F-central ZG-factors, then A has an F-decomposition; (2) if there exists an irreducible F-central submodule B of A such that all ZG-composition factors of A/B are F-ecentric, then A has an F-decomposition.展开更多
Let R be a commutative ring having nonzero identity and M be a unital R-module.Assume that S⊆R is a multiplicatively closed subset of R.Then,M satisfies S-Noetherian spectrum condition if for each submodule N of M,ther...Let R be a commutative ring having nonzero identity and M be a unital R-module.Assume that S⊆R is a multiplicatively closed subset of R.Then,M satisfies S-Noetherian spectrum condition if for each submodule N of M,there exist s∈S and afinitely generated submodule F⊆N such that sN⊆radM(F),where radM(F)is the prime radical of F in the sense(McCasland and Moore in Commun Algebra 19(5):1327–1341,1991).Besides giving many properties and characterizations of S-Noetherian spectrum condition,we prove an analogous result to Cohen’s theorem for modules satisfying S-Noetherian spectrum condition.Moreover,we characterize modules having Noetherian spectrum in terms of modules satisfying the S-Noetherian spectrum condition.展开更多
Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The followi...Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.展开更多
基金TheNationalNaturalScienceFoundationofChina (No .10 1710 74 )
文摘Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), f(p)≠ for each p∈π. The following conclutions are obtained: (1) if there exists a maximal submodule B of A such that A/B is F-central in G and B has no nonzero F-central ZG-factors, then A has an F-decomposition; (2) if there exists an irreducible F-central submodule B of A such that all ZG-composition factors of A/B are F-ecentric, then A has an F-decomposition.
文摘Let R be a commutative ring having nonzero identity and M be a unital R-module.Assume that S⊆R is a multiplicatively closed subset of R.Then,M satisfies S-Noetherian spectrum condition if for each submodule N of M,there exist s∈S and afinitely generated submodule F⊆N such that sN⊆radM(F),where radM(F)is the prime radical of F in the sense(McCasland and Moore in Commun Algebra 19(5):1327–1341,1991).Besides giving many properties and characterizations of S-Noetherian spectrum condition,we prove an analogous result to Cohen’s theorem for modules satisfying S-Noetherian spectrum condition.Moreover,we characterize modules having Noetherian spectrum in terms of modules satisfying the S-Noetherian spectrum condition.
文摘Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.
