Rings with S-Noetherian spectrum

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a multiplicative subset of [Formula: see text]. We say that [Formula: see text] has [Formula: see text]-Noetherian spectrum if for every ideal [Formula: see text] of [Formula: see text] [Formula: see text] for some [Formula: see text] and some finitely generated ideal [Formula: see text] In this paper, we study rings with [Formula: see text]-Noetherian spectrum. Among other things, we give a necessary and sufficient condition for Nagata’s idealization [Formula: see text], where [Formula: see text] is an [Formula: see text]-module, to satisfy the [Formula: see text]-Noetherian spectrum. We also investigate when the amalgamated algebra along an ideal has [Formula: see text]-Noetherian spectrum. Several examples are given to illustrate the concepts and results.