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

Let 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?^fJ), 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.