scholarly journals A Difference-Differential Basis Theorem

1970 ◽  
Vol 22 (6) ◽  
pp. 1224-1237 ◽  
Author(s):  
Richard M. Cohn
Keyword(s):  
Commutative Ring ◽  
Chain Condition ◽  
Finitely Generated ◽  
Hilbert Basis ◽  
Radical Ideal ◽  
Differential Basis ◽  
Basis Theorem ◽  
Finite Set ◽  
Radical Ideals ◽  
Ascending Chain Condition

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [5, Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained.What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R0, then the ascending chain condition for radical ideals of R0 implies the ascending chain condition for radical ideals of R. (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R.) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

scholarly journals On minimal log discrepancies and kollár components

2021 ◽  
pp. 1-20
Author(s):  
Joaquín Moraga
Keyword(s):  
Blow Up ◽  
Chain Condition ◽  
Fano Varieties ◽  
Finite Sets ◽  
Fixed Dimension ◽  
Log Canonical ◽  
Finite Set ◽  
Canonical Singularities ◽  
Ascending Chain Condition

Abstract In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

Noetherian orders

2010 ◽  
Vol 21 (1) ◽  
pp. 111-124 ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Noetherian Ring ◽  
Constructive Proof ◽  
Finitely Generated ◽  
Hilbert Basis ◽  
Basis Theorem ◽  
Natural Way

Noether classes of posets arise in a natural way from the constructively meaningful variants of the notion of a Noetherian ring. Using an axiomatic characterisation of a Noether class, we prove that if a poset belongs to a Noether class, then so does the poset of the finite descending chains. When applied to the poset of finitely generated ideals of a ring, this helps towards a unified constructive proof of the Hilbert basis theorem for all Noether classes.

Some results on finiteness of radical algebras

1970 ◽  
Vol 11 (3) ◽  
pp. 291-296
Author(s):  
Ahmad Mirbagheri
Keyword(s):  
Chain Condition ◽  
Finitely Generated ◽  
Radical Algebra ◽  
Finite Dimensional ◽  
Ascending Chain Condition ◽  
Algebra Over A Field

R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then (i) R is finitely generated (ii) R is left alehra noetherian (iii) R is algebraic. Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.

On a class of Noetherian algebras

1999 ◽  
Vol 129 (6) ◽  
pp. 1185-1196 ◽  
Author(s):  
E. Jespers ◽  
J. Okniński
Keyword(s):  
Finite Group ◽  
Homomorphic Image ◽  
Chain Condition ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Finitely Generated ◽  
Nilpotent Semigroup ◽  
Left And Right ◽  
Ascending Chain Condition

A class of Noetherian semigroup algebrasK[S]is described. In particular, we show that, for any submonoidSof the semigroupMnof all monomialn × nmatrices over a polycyclic-by-finite groupG, K[S]is right Noetherian if and only ifSsatisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroupSsatisfying the ascending chain condition on right ideals is left and right Noetherian.

scholarly journals Semigroups whose right ideals are finitely generated

2021 ◽  
pp. 1-36
Author(s):  
Craig Miller
Keyword(s):  
Sufficient Conditions ◽  
Chain Condition ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Equivalent Formulation ◽  
Regular Semigroups ◽  
Strong Semilattice ◽  
Necessary And Sufficient ◽  
Ascending Chain Condition ◽  
Completely Simple

Abstract We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

2020 ◽  
Vol 27 (03) ◽  
pp. 531-544
Author(s):  
Farid Kourki ◽  
Rachid Tribak
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Chain Condition ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Cyclic Submodules ◽  
Descending Chain Condition

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

Subrings of the Maximal Ring of Quotients Associated With Closure Operations

1963 ◽  
Vol 15 ◽  
pp. 723-743 ◽  
Author(s):  
D. C. Murdoch
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Local Ring ◽  
Complete Solution ◽  
Chain Condition ◽  
Central Problem ◽  
Closure Operations ◽  
Ring Of Quotients ◽  
Ascending Chain Condition

This paper contains a number of results that grew out of an attempt to solve the following problem : Given a non-commutative ring R with suitable ascending chain condition, and a prime ideal P in R, to construct a corresponding local ring RP in which the extension P′ of P is a unique maximal prime, and to prove, if possible, that the intersection of the powers of P′ is zero. The present investigation is at best a preliminary attack on this problem since the contribution to the complete solution is comparatively small and the central problem of the intersection of the powers of P′ has not been touched.

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