Cohen-Macaulay properties under the amalgamated construction

Abstract

Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then, the subring $A\bowtie ^fJ:=\{(a,f(a)+j)\mid a\in A$ and $j\in J\}$ of $A\times B$ 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 \cite {AT}, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $A\bowtie ^fJ$. Among other things, we obtain a generalization of the well-known result of when Nagata's idealization is Cohen-Macaulay.