On Modules Satisfying S-Noetherian Spectrum Condition

Abstract

Let R be a commutative ring having nonzero identity and M be a unital R-module. Assume that S subset of 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 is an element of S and a finitely generated submodule F subset of N such that sN subset of 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.