Amalgamated algebras along an ideal

Let $f : A \rightarrow B$ be a ring homomorphism and J an ideal of B. In this paper, we initiate a systematic study of a new ring construction called the “amalgamation of A with B along J with respect to f ”. This construction finds its roots in a paper by J.L. Dorroh appeared in 1932 and 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] and A + XB[[X]] constructions, the CPI-extensions of Boisen and Sheldon, the D + M constructions and the Nagata’s idealization.