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.

We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R,ℳ)$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) $ℳ={⨁}_{\lambda \in \Lambda }R{w}_{\lambda }$ where $\Lambda$ is an index set and $R/Ann\left(w_{\lambda }\right)$ is a principal ideal ring for each $\lambda \in \Lambda$; (3) Every prime ideal of $R$ is a direct sum of at most...

A module M satisfies the restricted minimum condition if M/N is artinian for every essential submodule N of M. A ring R is called a right RM-ring whenever $R_R$ satisfies the restricted minimum condition as a right module. We give several structural necessary conditions for particular classes of RM-rings. Furthermore, a commutative ring R is proved to be an RM-ring if and only if R/Soc(R) is noetherian and every singular module is semiartinian.

It is shown that for any Artinian modules $M$, $dim{M}^{\vee }$ is the greatest integer $i$ such that ${\text{H}}_{𝔪}^i\left(M^{\vee }\right)\ne 0$.

Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 1$ for all $i$, then the $R$-module $H_I^i\left(M\right)$ is $I$-cominimax for all $i$. In fact, $H_I^i\left(M\right)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 2$ for all $i<t$, then ${\mathrm{Ext}}_R^j(R/I,H_I^i\left(M\right))$ and ${\mathrm{Hom}}_R(R/I,H_I^t\left(M\right))$ are weakly Laskerian for all $i<t$ and all $j\ge 0$. As a consequence, the set of associated primes of $H_I^i\left(M\right)$ is finite for all $i\ge 0$, whenever $dimR/I\le 2$ and...