Noetherian Condition Research Articles

Applications of $\bigcap$-large pseudo N-injective acts in quasi-Frobenius monoid theory and its relationship with some classes of injectivity

The aim of this paper is to review thoroughly the applications of ∩ -large pseudo N-injective acts in quasi-Frobenius monoid theory, and therefore, the relationship of ∩ -large pseudo N-injective acts with some class of injectivity is studied. Applications of the properties of ∩ -large pseudo injective acts in quasi-Frobenius monoid theory are proven. Also, it’s proved that the subsequent parity, every union (direct sum) of the two ∩ -large pseudo injective acts is a ∩ -large pseudo injective act if and only if every ∩ -large pseudo injective act is injective under Noetherian condition for a right monoid S. Additionally, we proved that the category of strongly ∩ -large pseudo N-injective right S-acts are going to be egalitarian to the category of projective right S-acts under monoid conditions. The connections between quasi injective and ∩ -large pseudo injective acts are investigated.

Elementary preservation of the Noetherian condition and the potential applications in computational physics

We give an elementary proof of the preservation of the Noetherian condition for commutative rings with unity R having at least one finitely generated ideal I such that the quotient ring is again finitely generated, and R is I—adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. Furthermore, we discuss the potential applications that this new elementary proof possesses regarding the simplification of conceptual generations in mathematics and computational physics based on the new computational paradigm of Artificial Mathematical Intelligence. In addition, we give a counterexample showing that the ‘completion’ condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields.

Integer-valued skew polynomials

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

A generalized Noetherian condition for Lie algebras

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable.

The Noetherian Conditions and the Index of Some Class of Singular Integral Operators over a Bounded Simply Connected Domain

Necessary and sufficient conditions to be Noetherian are obtained for two-dimensional singular integral operators on Lebesgue spaces with a weight coefficient. A formula for calculation of their index is given.

Poor modules with no proper poor direct summands

As a mean to provide intrinsic characterizations of poor modules, the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand. We show that not all rings have pauper modules and explore conditions for their existence. In addition, we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity: one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule. The second condition holds for the ring of integers and is just as significant as the first one for Noetherian rings since, in that context, modules having poor pure submodules must themselves be poor.It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class. As indecomposable poor modules are pauper, we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective). We show that semiartinian V-rings satisfy this property and also that a commutative Noetherian ring R has no indecomposable middle class if and only if R is the direct product of finitely many fields and at most one ring of composition length 2. Structure theorems are also provided for rings without indecomposable middle class when the rings are Artinian serial or right Artinian.Rings for which not having an indecomposable middle class suffices not to have a middle class include commutative Noetherian and Artinian serial rings. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized.

Residue fields for a class of rational E∞-rings and applications

Let A be an E∞-ring over the rational numbers. If A satisfies a noetherian condition on its homotopy groups π⁎(A), we construct a collection of E∞-A-algebras that realize on homotopy the residue fields of π⁎(A). We prove an analog of the nilpotence theorem for these residue fields. As a result, we are able to give a complete algebraic description of the Galois theory of A and of the thick subcategories of perfect A-modules. We also obtain partial information on the Picard group of A.

Noetherian Algebras of Quantum Differential Operators

We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine the algebra of quantum differential operators as a skew group algebra of the group $\mathbb{Z}^n$ over a quantized Weyl algebra. It follows from this description that this algebra is a simple right and left Noetherian domain.

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.

ON THE FINITE TYPE OF FAMILIES OF INDECOMPOSABLE MODULES

We show that a family [Formula: see text] of indecomposable modules has only a finite number of non-isomorphic members if [Formula: see text] satisfies either of the following: (a) [Formula: see text] has a preinjective partition, and satisfies the artinian condition on morphisms between the Mi; (b) [Formula: see text] has a preprojective partition, and satisfies the noetherian condition on morphisms between the Mi. As a consequence, we recover some of recent results due to B. Huisgen-Zimmermann and M. Saorín [11], establishing the relationships between the endo-structure of infinite direct sums ⊕i∈IMi of indecomposable modules Mi and the finiteness of isomorphism classes of the Mi.

Bi-artinian noetherian rings

A noetherian ring R satisfies the descending chain condition on two-sided ideals (“is bi-artinian”) if and only if, for each prime P ∈ spec(R), R/P has a unique minimal ideal (necessarily idempotent and left-right essential in R/P). The analogous statement for merely right noetherian rings is false, although our proof does not use the full noetherian condition on both sides, requiring only that two-sided ideals be finitely generated on both sides and that R/Q be right Goldie for each Q ∈ spec(R). Examples exist, for each n∈ℕ and in all characteristics, of bi-artinian noetherian domains Dn with composition series of length 2n and with a unique maximal ideal of height n. Noetherian rings which satisfy the related E-restricted bi-d.c.c. do not, in general, satisfy the second layer condition (on either side), but do satisfy the Jacobson conjecture.

Noetherian Banach Jordan pairs

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

Noetherian Jordan Banach Algebras Are Finite-Dimensional

w x Sinclair and Tullo proved in 10 that Noetherian Banach algebras, that is, complex associative Banach algebras satisfying the ascending chain w x condition on left ideals, are finite-dimensional. In 4 , we extended this result to the alternative case. For a Jordan algebra, the natural Noetherian condition is the ascending chain condition on inner ideals. Here, we show that Noetherian Jordan Banach algebras are finite-dimensional.

On the Noetherian condition for implectic operators and bi-Hamiltonian structure

We have demonstrated in the case of the complex mKdV equation that the bi-Hamiltonian structure can be deduced by solving the Noetherian condition to be satisfied by the implectic operators. The technique of solving the Noetherian equation is perturbative. The strong and hereditary condition is seen to be satisfied by the recursion operator constructed from the two implectic operators.

Two notes on mutiplicatively closed sets of ideals and divisors

LetΓ be a mulitiplicatively closed set of nonzero idcals of a Noetherian ring R and for an ideal I of R let IΓ=∪{I:G;G∈Γlcub; Then IΓ is an ideal in R and it is shown that if P is prime divisor of IΓ, then P is a prime divisor of IG:G′ for all G,G′ In $GgR;. Alao, for a given filtration is a filtration on R and when φ is Noetherian several necessary and sufficient conditions are given for φ and φΓ to give linearly equivalent ideal topologies on R .

An elementary derivation of an inequality involvingR-sequences

A basic inequality is derived for the grade of a finitely generated ideal on a general module. The methods used are both simple and elementary. No Noetherian conditions are needed.

Dual generalizations of the Artinian and Noetherian conditions

An Artίnίan module can be characterized in terms of certain properties of its factor modules. A module Mis Artinian if and only if the following two conditions hold for M: (I) Every nonzero factor module of M contains a minimal submodule. (A) The socle of every factor module ofM is finitely generated. The dual to the factor module is the submodule. We state the dual of (I): (Π) Every nonzero submodule ofM contains a maximal submodule. We call a module with property (Π) a Max module and one with property (I) a Min module. Every Noetherian module is a Max module but not conversely. This paper investigates these generalizations of the Artinian and Noetherian conditions and the relationships among them. Throughout this paper M denotes a right module over an arbitrary

Noetherian and artinian chain conditions of associative rings

Noetherian and artinian chain conditions of associative rings