The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex $Ho{m}_{R̂}((,R̂),M)$ is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.

In this paper, we use a characterization of $R$-modules $N$ such that $f{d}_{R}N=p{d}_{R}N$ to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting $N$ to be the $dth$ local cohomology functor of $R$ with respect to the maximal ideal where $d$ is the Krull dimension of $R$.

Let $𝔞$, $I$, $J$ be ideals of a Noetherian local ring $(R,𝔪,k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I,J}^t\left(M\right)$ and $D\left(H_{I,J}^t\left(M\right)\right)$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):={\mathrm{Hom}}_R(-,E_R\left(k\right))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H_{I,J}^i\left(R\right)=0$...

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 show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C{\otimes }_{R}M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.

We obtain, in a simple way, an estimate for the Noether exponent of an ideal I without embedded components (i.e. we estimate the smallest number μ such that ${\left(radI\right)}^{\mu }\subset I$).

Let $K$ be a field and $S=K[x_1,...,x_m,y_1,...,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” $S$-modules with respect to $Q=(y_1,...,y_n)$. Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to $Q$ are considered.

In this note we give a description of a morphism related to the structure of the canonical model of the Rees algebra R(I) of an ideal I in a local ring. As an application we obtain Ikeda's criteria for the Gorensteinness of R(I) and a result of Herzog-Simis-Vasconcelos characterizing when the canonical module of R(I) has the expected form.