Rings with S-Noetherian spectrum

Let R be a commutative ring with identity and S be a multiplicative subset of R. We say that R has S-Noetherian spectrum if for every ideal I of R, sI subset of root J subset of root I for some s is an element of S and some finitely generated ideal J. In this paper, we study rings with S-Noetherian spectrum. Among other things, we give a necessary and sufficient condition for Nagata's idealization R(+)M, where M is an R-module, to satisfy the S-Noetherian spectrum. We also investigate when the amalgamated algebra along an ideal has S-Noetherian spectrum. Several examples are given to illustrate the concepts and results.