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 prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...

Recall that a regular local ring is a noetherian local ring with dimension equal to dimkm/m. A regular local ring of dimension one is precisely a discrete valuation ring. If V is a nosingular variety then t(V ) is a scheme with all local rings regular, so t(V ) is clearly regular in codimension one. In this section we will consider schemes satisfying the following condition: (∗) X is a noetheri...

for each  and i ≥ 0. The polynomial ring of integer-valued in rational  is defined by Int (    an important example binomial and is non-Noetherian ring. In this paper the algebraic structure rings has been studied their properties ideals. notion ideal generated a given set defined.  Which allows us to define new class Noetherian using ideals, which we named it binomiall...

Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

For every local ring R in what follows, mR denotes the maximal ideal. Let Λ̃ be a complete, regular, local Noetherian ring, let E ⊂ m Λ̃ be an ideal, and denote the quotient by Λ. Thus, Λ is also a complete, local Noetherian ring. Denote the residue field Λ/mΛ by k. Denote by C = CΛ the category whose objects are Λ-algebras A such that (i) A is a local, Artin ring with mΛA ⊂ mA, and (ii) the indu...

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially “sewn together” to form SpecR. In particular, we construct a bimodule-determined functor ModZ → ModR, for a suitable commutative noetherian ring Z, from which there...