An R-module M is said to have uniformly S-Noetherian spectrum, where \(S\subseteq R\) is a multiplicatively closed set, if there exists \(s\in S\) such that for every submodule H of M, \(sH\subseteq rad_{M}(K)\) for some finitely generated submodule K of H. Cohen’s theorem and Eakin-Nagata-Formanek theorem are provided for modules having uniformly S-Noetherian spectrum. In addition, an analogous result to the Hilbert basis theorem is given for rings having uniformly S-Noetherian spectrum.
