Rings with \(S\)-Noetherian spectrum (Q6607145)

Let \(R\) be a commutative ring with identity and \(S\) be a multiplicative subset of \(R\). \(R\) is said to have \(S\)-Noetherian spectrum if for every ideal \(I\) of \(R\), \(sI\subseteq \sqrt{J}\subseteq \sqrt{I}\) for some \(s\in S\) and some finitely generated ideal \(J\). In analogy to results on Noetherian rings, Ohm and Pendleton proved many results including Cohen's theorem and the Hilbert basis theorem for rings with Noetherian spectrum. More precisely, they showed that \(R\) has Noetherian spectrum if and only if every prime ideal of \(R\) is radically finite; if \(R\) has Noetherian spectrum, then so does the polynomial ring \(R[X]\). In this paper, authors study rings with \(S\)-Noetherian spectrum. In Section 1, they studied a relation between Noetherian spectrum and \(S\)-Noetherian spectrum. More precisely, they constructed an example of a commutative ring \(R\) and a multiplicative subset \(S\) of \(R\) such that \(R\) has S-Noetherian spectrum, but does not have Noetherian spectrum. Further they provided a new characterization of rings with Noetherian spectrum in terms of rings with \(S\)-Noetherian spectrum. In Section 2, authors provide a necessary and sufficient condition for Serre's conjecture ring and the Nagata ring to satisfy the S-Noetherian spectrum property. In Section 3, they gave an equivalent condition for the Nagata (respectively, \(t\)-Nagata) ring to satisfy the locally (respectively, \(t\)-locally) \(S\)-Noetherian spectrum property. Finally they provided a necessary and sufficient condition for Nagata's idealization \(R(+)M\), where \(M\) is an \(R\)-module, to satisfy the \(S\)-Noetherian spectrum. They also investigated when the amalgamated algebra along an ideal has \(S\)-Noetherian spectrum.