Amalgamated algebra along an ideal defined by \(S\)-Noetherian spectrum-like-conditions (Q6600957)

A ring is said to have Noetherian spectrum if it satisfies the ascending chain conditions (ACC) on radical ideals. Note that every Noetherian ring has Noetherian spectrum. This notion was generalized by introducing the concept of \(S\)-Noetherian rings. Let \(R\) be a commutative ring with identity and \(S\) be a multiplicative subset of \(R.\) \(R\) is said to have a \(S\)-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 this paper, authors pursue the investigation on the structure of the rings of the form \(A(+)M\) and the amalgamated duplication with a particular attention to the \(S\)-Noetherian spectrum property. More precisely, they gave a necessary and sufficient condition for the trivial ring extension to satisfy the \(S(+)N\)-Noetherian spectrum property, where \(S\) is a multiplicative subset of \(R\) and \(N\) a submodule of the \(R\)-module \(M.\) In the last part of the paper, authors obtained a necessary and sufficient condition for amalgamation to satisfy the \(S^\prime\)-Noetherian spectrum.