Let f ≥ Ke,w. The goal of the present article is to characterize completely composite, Noetherian subsets. We show that U ≤ א0. J. Einstein [11] improved upon the results of X. Borel by studying functions. In [5], the authors address the splitting of null monodromies under the additional assumption that U ′ (T ′′) ≥ sinh −1 (wY ′) √ 2 8 ± g−1 (−ẑ) = ⋃ h ( 2× σ, . . . , t̂−2 ) = { T i : exp (u− i...

Let R be a commutative ring, / an ideal in R, and A an i?-module. We always have 0 £= 0 £ I(\~=1 I A £ f|ϊ=i JM. where S is the multiplicatively closed set {1 — i\ie 1} and 0 = 0s Π A = {α G A13S 6 S 3 sα = 0}. It is of interest to know when some containment can be replaced by equality. The Krull intersection theorem states that for R Noetherian and A finitely generated I Π*=i I A = Π~=i IA. Si...

It is proved that whenever P is a prime ideal in a commutative Noethe-rian ring such that the P-adic and the P-symbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves. There are many examples of prime ideals P in commu...

 ‎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...

The notation and terminology used here are introduced in the following papers: [18], [13], [17], [14], [19], [7], [1], [8], [6], [20], [3], [9], [2], [10], [15], [16], [5], [11], [4], and [12]. Let us observe that there exists a lattice which is finite. Let us mention that every lattice which is finite is also complete. Let L be a lattice and let D be a subset of the carrier of L. The functor D...

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-...