Chain conditions in amalgamated algebras along an ideal (Q384555)

Let \(R\) be a commutative ring with unity. Recall that \(R\) is said to be nonnil-Noetherian if each ideal of \(R\) which is not contained in the nilradical of \(R\) is finitely generated. The ring \(R\) has Noetherian spectrum if it satisfies the ascending chain condition for radical ideals. Now, let \(A\) and \(B\) be two rings, \(J\) an ideal of \(B\) and \(f:A\longrightarrow B\) be a ring homomorphism. We consider the following subring \(\{(a,f(a)+j);~a\in A, j\in J\}\) of \(A\times B\), called the amalgamation of \(A\) with \(B\) along \(J\) with respect to \(f\). In this paper, the authors characterize when the amalgamation is nonnil-Noetherian and when it has a Noetherian spectrum.