Properties of chains of prime ideals in an amalgamated algebra along an ideal

Abstract

a b s t r a c tLet f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A f J), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions (such as the A + XB[X ], the A + XB[[X ]] and the D + M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.

On Almost Valuation and Almost Bézout Rings

“…In the sequel, we assume that J is a nonzero proper ideal of B, and we denote by Nilp B the set of all nilpotent elements of B. The proof of this theorem requires the following preparatory lemma which is an immediate consequence of [11,Proposition 2.1 (3)] and the fact that if A is an AV -ring and I is an ideal of A, then A/I is an AV -ring.…”

“…In this setting, we consider the subring of A × B, A f J = a f a + j a ∈ A j ∈ J called the amalgamation of A and B along J with respect to f . This construction is a generalization of the amalgamated duplication of a ring along an ideal (introduced and studied by D'Anna and Fontana in [9,12,13] [10][11][12][13]. Unreferenced material is standard as in [15].…”

Global dimension of bi-amalgamated algebras along pure ideals

“…Given a ring homomorphism f : A → B and an ideal J of B, the bi-amalgamation A ι,f (f −1 (J), J) coincides with the amalgamated algebra introduced in 2009 by D'Anna, Finocchiaro, and Fontana [7,8] as the following subring of A × B:…”

Bi-Amalgamation of Small Weak Global Dimension

“…This construction was introduced in [13] as a natural generalization of duplications [5,6] and amalgamations [7,8]. Given a ring homomorphism f : A → B and an ideal …”