Properties of chains of prime ideals in an amalgamated algebra along an ideal (Q964519)

Let \(A\) and \(B\) be commutative rings with unity, let \(J\) be an ideal of \(B\) and let \(f:A \to B\) be a ring homomorphism. Consider the following subring of \(A \times B\): \({A\bowtie^fJ}:= \{(a, f(a) +j)\mid a \in A, j \in J\}\) called the amalgamation of \(A\) with \(B\) along \(J\) with respect to \(f\). In this paper, the authors study this amalgamation, which is a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, which was introduced and studied by the first and third author [J. Algebra Appl. 6, No. 3, 443--459 (2007; Zbl 1126.13002); Ark. Mat. 45, No. 2, 241--252 (2007; Zbl 1143.13002)], and other classical constructions (such as the \(A+ XB[X]\), the \(A+ XB[[X]]\) and the \(D+M\) constructions). In particular, the authors completely described the prime spectrum of the amalgamated duplication and they give bounds for its Krull dimension.