Cohen-Macaulay properties under the amalgamated construction

Let $A$ and $B$ be commutative rings with unity, $f:Ato B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $Abowtie^fJ:={(a,f(a)+j)|ain A$ and $jin J}$ of $Atimes B$ 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, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $Abowtie^fJ$. Among other things, we obtain a generalization of the well-known result that when the Nagatas idealization is Cohen-Macaulay.