Decomposition of ideals into pseudo-irreducible ideals in amalgamated algebra along an ideal

Let f:ArightarrowB be a ring homomorphism and J an ideal of B. In this paper, we give a necessary and sufficient condition for the amalgamated algebra along an ideal AbowtiefJ to be J-Noetherian. Then we give a characterization for pseudo-irreducible ideals of AbowtiefJ, in special cases.