New Algebraic Properties of an Amalgamated Algebra Along an Ideal

Let f: A → B be a ring homomorphism, and let J be an ideal of B. In this article, 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 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 amalgamation A ⋈fJ and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen–Macaulay ring or a Gorenstein ring.