scholarly journals Noetherian and Artinian ordered groupoids—semigroups

2005 ◽  
Vol 2005 (13) ◽  
pp. 2041-2051
Author(s):  
Niovi Kehayopulu ◽  
Michael Tsingelis
Keyword(s):  
Chain Condition ◽  
Growth Conditions ◽  
Ideal Theory ◽  
Minimum Condition ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Maximal Ideals ◽  
Ordered Groupoid ◽  
Descending Chain Condition

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (semigroups) which are about to be introduced are particularly versatile. These satisfy a certain finiteness condition, namely, that every ideal of the ordered groupoid (semigroup) is finitely generated. Our purpose is to introduce the concepts of Noetherian and Artinian ordered groupoids. An ordered groupoid is said to be Noetherian if every ideal of it is finitely generated. In this paper, we prove that an equivalent formulation of the Noetherian requirement is that the ideals of the ordered groupoid satisfy the so-called ascending chain condition. From this idea, we are led in a natural way to consider a number of results relevant to ordered groupoids with descending chain condition for ideals. We moreover prove that an ordered groupoid is Noetherian if and only if it satisfies the maximum condition for ideals and it is Artinian if and only if it satisfies the minimum condition for ideals. In addition, we prove that there is a homomorphismπof an ordered groupoid (semigroup)Shaving an idealIonto the Rees quotient ordered groupoid (semigroup)S/I. As a consequence, ifSis an ordered groupoid andIan ideal ofSsuch that bothIand the quotient groupoidS/Iare Noetherian (Artinian), then so isS. Finally, we give conditions under which the proper prime ideals of commutative Artinian ordered semigroups are maximal ideals.

scholarly journals A note on universally zero-divisor rings

1991 ◽  
Vol 43 (2) ◽  
pp. 233-239 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Integral Extension ◽  
Maximal Ideals ◽  
Unitary Module ◽  
Finite Set ◽  
Descending Chain Condition

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

scholarly journals Bi-artinian noetherian rings

2001 ◽  
Vol 43 (1) ◽  
pp. 9-21
Author(s):  
E. A. Whelan
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Ideal ◽  
Chain Condition ◽  
Noetherian Rings ◽  
Composition Series ◽  
Finitely Generated ◽  
Analogous Statement ◽  
Descending Chain Condition ◽  
Unique Maximal Ideal

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.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

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.

On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

2009 ◽  
Vol 30 (6) ◽  
pp. 1803-1816 ◽  
Author(s):  
C. R. E. RAJA
Keyword(s):  
London Math ◽  
Abelian Groups ◽  
Chain Condition ◽  
Compact Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Metrizable Group ◽  
Compact Abelian Groups ◽  
Descending Chain Condition

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

scholarly journals Trees as commutative BCK-Algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 181-190
Author(s):  
William H. Cornish
Keyword(s):  
Height Function ◽  
Chain Condition ◽  
New Method ◽  
Subdirectly Irreducible ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Congruence Distributivity ◽  
Finite Height ◽  
Descending Chain Condition ◽  
Simple Algebras

A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

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.

scholarly journals The descending chain condition on solution sets for systems of equations in groups

1986 ◽  
Vol 29 (1) ◽  
pp. 69-73 ◽  
Author(s):  
M. H. Albert ◽  
J. Lawrence
Keyword(s):  
Free Group ◽  
Nonempty Subset ◽  
Free Monoid ◽  
Chain Condition ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Finitely Generated ◽  
Solution Sets ◽  
Number Of Generators ◽  
Descending Chain Condition

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions

1975 ◽  
Vol 27 (1) ◽  
pp. 75-87 ◽  
Author(s):  
Andrew Adler ◽  
R. Douglas Williams
Keyword(s):  
Entire Functions ◽  
Continuous Functions ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Maximal Ideals ◽  
Primary Ideals ◽  
Special Case ◽  
The Ideal

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.

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