GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

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.

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MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES

Journal of Algebra and Its Applications ◽

10.1142/s0219498804000939 ◽

2004 ◽

Vol 03 (04) ◽

pp. 391-410 ◽

Cited By ~ 4

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074697 ◽

1996 ◽

Vol 120 (1) ◽

pp. 87-101 ◽

Cited By ~ 13

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].

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BOUNDED AND FULLY BOUNDED MODULES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002814 ◽

2011 ◽

Vol 84 (3) ◽

pp. 433-440

Author(s):

A. HAGHANY ◽

M. MAZROOEI ◽

M. R. VEDADI

Keyword(s):

Krull Dimension ◽

Finitely Generated ◽

Morita Invariant ◽

Prime Submodule ◽

Invariant Properties ◽

Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

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Modular C11 lattices and lattice preradicals

Journal of Algebra and Its Applications ◽

10.1142/s021949881750116x ◽

2017 ◽

Vol 16 (06) ◽

pp. 1750116 ◽

Cited By ~ 1

Author(s):

Toma Albu ◽

Mihai Iosif

Keyword(s):

Modular Lattices ◽

Torsion Theories

This paper deals with properties of modular [Formula: see text] lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

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LOCALIZATION OF MODULAR LATTICES, KRULL DIMENSION, AND THE HOPKINS-LEVITZKI THEOREM. II

Communications in Algebra ◽

10.1081/agb-100105048 ◽

2001 ◽

Vol 29 (8) ◽

pp. 3677-3682 ◽

Cited By ~ 3

Author(s):

Toma Albu ◽

Patrick F. Smith

Keyword(s):

Krull Dimension ◽

Modular Lattices

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On a common abstraction of de Morgan algebras and Stone algebras

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015663 ◽

1983 ◽

Vol 94 (3-4) ◽

pp. 301-308 ◽

Cited By ~ 33

Author(s):

T. S. Blyth ◽

J. C. Varlet

Keyword(s):

Distributive Lattice ◽

Subdirectly Irreducible ◽

Bounded Distributive Lattice ◽

Subdirect Irreducibility ◽

De Morgan Algebras ◽

Simple Equations ◽

Lattice Of Congruences

SynopsisWe consider a common abstraction of de Morgan algebras and Stone algebras which we call an MS-algebra. The variety of MS-algebras is easily described by adjoining only three simple equations to the axioms for a bounded distributive lattice. We first investigate the elementary properties of these algebras, then we characterise the least congruence which collapses all the elements of an ideal, and those ideals which are congruence kernels. We introduce a congruence which is similar to the Glivenko congruence in a p-algebra and show that the location of this congruence in the lattice of congruences is closely related to the subdirect irreducibility of the algebra. Finally, we give a complete description of the subdirectly irreducible MS-algebras.

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Torsion theoretic points and spaces

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500025476 ◽

1984 ◽

Vol 96 (3-4) ◽

pp. 345-361 ◽

Cited By ~ 10

Author(s):

Harold Simmons

Keyword(s):

Topological Space ◽

Krull Dimension ◽

Noetherian Rings ◽

Decomposition Theory ◽

Torsion Theories ◽

Natural Decomposition

SynopsisFor each ring R, we construct a topological space pt (R) which includes as a subspace both the classical spectrum specR and the torsion theoretic spectrum R-sp. For many rings (e.g. rings with Krull dimension), spec R is a retract of pt (R) and the retraction map θ generalizes the Gabriel correspondence for noetherian rings. There is a natural decomposition theory on MOD-R which extends the Goldman theory in the same way that the tertiary theory extends the primary theory. The map θ provides a direct comparison between this new decomposition theory and the tertiary theory. The space pt (R) is closely connected with the lattice of hereditary torsion theories on R, and for fully bounded (not necessarily noetherian) R, this connection is very tight.

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Goldie dimensions of quotient modules

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002688 ◽

2001 ◽

Vol 71 (1) ◽

pp. 11-19

Author(s):

John Dauns

Keyword(s):

Associative Ring ◽

Infinite Cardinal ◽

Goldie Dimension ◽

Injective Modules ◽

Subdirect Products ◽

Quotient Modules ◽

Quotient Property

AbstractFor an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

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