On Modules Satisfying S-Noetherian Spectrum Condition

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 a finitely generated submodule F ⊆ N such that s N ⊆ rad M ( F ) , where rad M ( 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.