et  be a commutative Noetherian ring,  and  two ideals of  and  a finite -module. In this paper, we have studied the vanishing and relative Cohen-Macaulyness of the functor for relative Cohen-Macauly filtered modules with respect to the ideal  (RCMF). We have shown that the for relative Cohen-Macaulay modules holds for any relative Cohen-Macauly module with respect to  with ........

We introduce and investigate the notion of GC -projective modules over (possibly non-noetherian) commutative rings, where C is a semidualizing module. This extends Holm and Jørgensen’s notion of C-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite GC-projectiv...

Let R be a commutative noetherian ring. We study the behavior of injectiveand at dimension of R-modules under the functors HomR(-,-) and -×R-.

Let $G$ be an abelian group and $S$ a given multiplicatively closed subset of commutative $G$-graded ring $A$ consisting homogeneous elements. In this paper, we introduce study $S$-Noetherian modules which are generalization modules. We characterize in terms For instance, $A$-module $M$ is if only $S$-Noetherian, provided finitely generated countable. Also, generalize some results on Noetherian...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.