Chain conditions in amalgamated algebras along an ideal

Let A and B be two rings, let J be an ideal of B and let f : A → B be a ring homomorphism. In this paper, we study when the amalgamation of A with B along J with respect to f is a $${\phi}$$-ring. Hence, we study two different chain conditions over this structure. Namely, the nonnil-Noetherian condition and the Noetherian spectrum condition.

constructions (such as the A+ X B [X ], A+ X B [[X ]], and the D + M constructions) can be studied as particular cases of the amalgamation [11, Examples 2.5 and 2.6] and other classical constructions, such as the Nagata's idealization, cf. [17, page 2], and the CPI extensions (in the sense of Boisen and Sheldon [7]) are strictly related to it [11,Example 2.7 and Remark 2.8].
On the other hand, the amalgamation A f J is related to a construction proposed by Anderson in [1] and motivated by a classical construction due to Dorroh [14], concerning the embedding of a ring without identity in a ring with identity. An ample introduction on the genesis of the notion of amalgamation is given in [11,Section 2]. Also, the authors consider the iteration of the amalgamation process, giving some geometrical applications of it.
One of the key tools for studying A f J is based on the fact that the amalgamation can be studied in the frame of pullback constructions [11,Section 4]. This point of view allows the authors in [11] and [12] to provide an ample description of various properties of A f J , in connection with the properties of A, J and f . Namely, in [11], the authors studied the basic properties of this construction (e.g., characterizations for A f J to be a Noetherian ring, an integral domain, a reduced ring) and they characterized those distinguished pullbacks that can be expressed as an amalgamation. Moreover, in [12], they pursue the investigation on the structure of the rings of the form A f J , with particular attention to the prime spectrum, to the chain properties and to the Krull dimension.
Recall from [3] and [13] that a prime ideal of R is called a divided prime ideal if P ⊆ (x) for every x ∈ R P; thus a divided prime ideal is comparable to every ideal of R. In [2], [4] and [5], the author paid attention to the class of rings Let A and B be two rings, let J be an ideal of B and let f : A → B be a ring homomorphism. In this paper, we study when the amalgamation of A with B along J with respect to f is a φ-ring.
Recall that a ring 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 treatment of this notion in the context of the class of φ-rings was established in [6], where the author proved that many of the properties of Noetherian rings are true for nonnil-Noetherian rings. Trivially, Noetherian rings are nonnil-Noetherian but the converse is not true in general, cf. [6,Theorem 3.4]. In Sect. 2, we characterize when A f J is nonnil-Noetherian provided it is φ-ring. Recall that a ring R has Noetherian spectrum if it satisfies the ascending chain condition for radical ideals. Every nonnil-Noetherian ring has Noetherian spectrum and the converse is false; cf. [16, Proposition 1.8 and Remark 1.9]. In Sect. 2, we characterize when A f J is of Noetherian spectrum.

Main results
We begin with the following result in which we study when A f J is a φ-ring.

Theorem 2.1 Let A and B be two rings, J be an ideal of B and let f : A → B be a ring homomorphism. If
A f J is a φ-ring then the following properties hold:  , f (a)). Hence, Let Recall that if A = B, f = id A and J is an ideal of A, the ring A id A J coincides with the amalgamated duplication of A along the ideal J defined in [10], as follows: Proof The result follows immediately by applying [15,Corollary 1.6], keeping in mind the fiber product structure of A f J = π • f × π where π is the canonical surjection f (A) + J → ( f (A) + J )/J , the fact that ( f (A) + J )/J is isomorphic to A f J/ f −1 (J ) × J and the fact that every subspace of a Noetherian topological space is still Noetherian.
We have the following consequence of the previous proposition.