<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

Let R be a commutative noetherian ring. Denote by $${\textsf{mod }}\,R$$ the category of finitely generated R-modules. In this paper, we study n-torsionfree modules in sense Auslander and Bridger, comparing them with n-syzygy modules, satisfying Serre’s condition $$(\mathrm {S}_n)$$ . We mainly investigate closedness properties full subcategories consisting those modules.

in this paper we introduce the notion of classical quasi-primary submodules that generalizes the concept of classical primary submodules. then, we investigate decomposition and minimal decomposition into classical quasi-primary submodules. in particular, existence and uniqueness of classical quasi-primary decompositions in finitely generated modules over noetherian rings are proved. moreover, w...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

Let $C$ be a semidualizing module over commutative Noetherian local ring $R$. In this paper we introduce new class of modules, namely $C$-canonical modules which are generalization canonical modules. It is shown that if the exists then and converse holds under special conditions. Also, characterization Gorenstein rings given via