COHEN-MACAULAY PROPERTIES UNDER THE AMALGAMATED CONSTRUCTION

Let A and B be commutative rings with unity, f : A -> B a ring homomorphism and J an ideal of B. Then, the subring A (sic)(f) J := {(a, f (a)+ j)vertical bar a is an element of A and j is an element of J} of A x B is called the amalgamation of A with B along J with respect to f. In this paper, we study the property of Cohen-Macaulay in the sense of ideals, which was introduced by Asgharzadeh and Tousi [2], a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring A (sic)(f) J. Among other things, we obtain a generalization of the well-known result of when Nagata's idealization is Cohen-Macaulay.