Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I)

1996 ◽  
Vol 120 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Toma Albu ◽  
Patrick F. Smith
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Torsion Theory ◽  
Krull Dimension ◽  
Main Body ◽  
Modular Lattices ◽  
Upper Continuous ◽  
Descending Chain Condition ◽  
Unital Rings

The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Năstăsescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].

scholarly journals GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

2010 ◽  
Vol 52 (A) ◽  
pp. 19-32 ◽  
Author(s):  
TOMA ALBU
Keyword(s):  
Krull Dimension ◽  
Subdirectly Irreducible ◽  
Modular Lattices ◽  
Survey Paper ◽  
Goldie Dimension ◽  
Subdirect Irreducibility ◽  
Upper Continuous ◽  
Torsion Theories ◽  
Quotient Modules ◽  
Noetherian Modules

AbstractIn this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

scholarly journals The descending chain condition in modular lattices

1972 ◽  
Vol 14 (4) ◽  
pp. 443-444
Author(s):  
Thomas G. Newman
Keyword(s):  
Elementary Proof ◽  
Chain Condition ◽  
Modular Lattices ◽  
Descending Chain Condition

In a recent paper Kovács [1] studied join-continuous modular lattices which satisfy the following conditions: (i) every element is a join of finitely many join-irredicibles, and, (ii) the set of join-irreducibles satisfies the descending chain condition. He was able to prove that such a lattice must itself satisfy the descending chain condition. Interest was expressed in whether or not one could obtain the same result without the assumption of modularity and/or of join-continuity. In this paper we give an elementary proof of this result without the assumption of join- continuity (which of course must then follow as a consequence of the descending chain condition). In addition we give a suitable example to show that modularity may not be omitted in general.

Chain Conditions for Modular Lattices with Finite Group Actions

1979 ◽  
Vol 31 (3) ◽  
pp. 558-564 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Modular Lattice ◽  
Chain Condition ◽  
Common Fixed Points ◽  
Finite Subgroup ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Combinatorial Result ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

scholarly journals Some characterizations of hereditarily artinian rings

1986 ◽  
Vol 28 (1) ◽  
pp. 21-23 ◽  
Author(s):  
Dinh van Huynh
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Descending Chain Condition ◽  
Associative Rings

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES

2004 ◽  
Vol 03 (04) ◽  
pp. 391-410 ◽  
Author(s):  
TOMA ALBU ◽  
MIHAI IOSIF ◽  
MARK L. TEPLY
Keyword(s):  
Modular Lattice ◽  
Independent Set ◽  
Torsion Theory ◽  
Modular Lattices ◽  
Finite Dimensional ◽  
Upper Continuous ◽  
Gabriel Dimension ◽  
Torsion Theories ◽  
Condition C ◽  
Compactly Generated

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

scholarly journals Equationally Compact Artinian Rings

1973 ◽  
Vol 25 (2) ◽  
pp. 273-283 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Rings ◽  
Direct Sums ◽  
Artinian Rings ◽  
Chain Conditions ◽  
Descending Chain Condition ◽  
Equational Compactness ◽  
Theoretical Results

By a Noetherian (Artinian) ring=(R;+ , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ringis said to beequationally compactif every system of ring polynomial equations with constants inis simultaneously solvable inprovided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

scholarly journals The descending chain condition in join-continuous modular lattices

1969 ◽  
Vol 10 (1-2) ◽  
pp. 1-4 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Chain Condition ◽  
Modular Lattices ◽  
Algebraic Systems ◽  
The Third ◽  
Irreducible Elements ◽  
Descending Chain Condition ◽  
A Chain

If L is a distributive lattice in which every element is the join of finitely many join-irreducible elements, and if the set of join-irreducible elements of L satisfies the descending chain condition, then L satisfies the descending chain condition: this follows easily from the results of Chapter VIII, Section 2, in the Third (New) Edition of Garrett Birkhoff's ‘Lattice Theory’ (Amer. Math. Soc., Providence, 1967). Certain investigations (M. S. Brooks, R. A. Bryce, unpublished) on the lattice of all subvarieties of some variety of algebraic systems require a similar result without the assumption of distributivity. Such a lattice is always join-continuous: that is, it is complete and (∧X) ∨ y = ∧ {x ∨ y: x ∈ X} whenever X is a chain in the lattice (for, the dual of such a lattice is complete and ‘algebraic’, in Birkhoff's terminology). The purpose of this note is to present the result:

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.

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