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
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.
