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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.
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 with $J$ with respect
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditi
For a partition $lambda$ of $n in {mathbb N}$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1,ldots,x_n]$ generated by all Specht polynomials of shape $lambda$. In the previous paper, the second author showed that if $R/I^{rm Sp}_lambda$ is Cohen-Ma
Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In particular, we sh
Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshornes connectedness theorem says that if (the coordina