Commutative Rings with Weakly S -Noetherian Spectrum

Let R be a commutative ring with identity and let S be a multiplicative subset of R. We say that R has weakly S-Noetherian spectrum if every ideal contained in a radically S-finite ideal is itself radically S-finite. In this paper, we introduce this notion and develop its basic properties. We prove several analogues of classical results for rings with S-Noetherian spectrum. In particular, we establish a Cohen-type result for the weakly S-Noetherian spectrum property. We also study the transfer of this property under localization and several related ring extensions.