‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...

In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely generated. (2) Even if a submodule of a finitely generated module is finitely generated, the minim...

in this paper we study some results on noetherian semigroups. we  show that if  $s_s$ is an  strongly  faithful $s$-act and $s$ is a duo weakly noetherian, then we have the following.

Let k be a field. We show that a finitely generated simple Goldie k-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and l...

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...

A differential ring of analytic functions in several complex variables is called a ring of Noetherian functions if it is finitely generated as a ring and contains the ring of all polynomials. In this paper, we give an effective bound on the multiplicity of an isolated solution of a system of n equations fi = 0 where fi belong to a ring of Noetherian functions in n complex variables. In the one-...

In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion i-Noetherian lattice introduced and related properties are investigated. We proved that iff every ideal principal. Moreover, show has spectrum it i-Noetherian.

Definition 2.1. A commutative ring R is Noetherian if every chain of ideals in R I0 ⊂ I1 ⊂ I2 ⊂ · · · terminates after a finite number of steps (i.e., there is an interger k such that Is = Ik if s ≥ k). Remark 2.2. Polynomial rings R [X1, . . . , Xn] are Noetherian if R is. In particular, S (p) = C [p∗] is Noetherian. Theorem 2.3. If R is a Noetherian ring, and M is a finitely generated R-modul...

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that t...