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 $S\subseteq R$ is a multiplicatively closed subset of R. Then, M satisfies S-Noetherian spectrum condition if for each submodule N of M, there exist $s\in S$ and a finitely generated submodule $F\subseteq N$ such that $sN\subseteq \text {rad}_{M}(F)$, where $\text {rad}_{M}(F)$ 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.

Keywords

Noetherian modules / S-Noetherian modules / Noetherian spectrum / S-Noetherian spectrum condition