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.

The conditions (Ci) in modular lattices, and applications

2015 ◽  
Vol 15 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Toma Albu ◽  
Mihai Iosif ◽  
Adnan Tercan
Keyword(s):  
Torsion Theory ◽  
Modular Lattices ◽  
Closed Element ◽  
Condition C

In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

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.

Torsion Theories Induced by Tilting Modules

1984 ◽  
Vol 36 (5) ◽  
pp. 899-913 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Torsion Theory ◽  
Torsion Class ◽  
Tilting Module ◽  
Tilting Modules ◽  
Direct Sums ◽  
Commutative Field ◽  
Finite Dimensional ◽  
Torsion Theories ◽  
Image Position ◽  
Torsion Free

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequencewith T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories andof the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.

An extension of F. Šik’s theorem on modular lattices

2018 ◽  
Vol 68 (6) ◽  
pp. 1321-1326
Author(s):  
Marcin Łazarz
Keyword(s):  
Finite Length ◽  
Modular Lattice ◽  
Unpublished Manuscript ◽  
Modular Lattices ◽  
Locally Finite ◽  
Semimodular Lattices ◽  
Upper Continuous ◽  
Atomic Lattices

AbstractJ. Jakubík noted in [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic toS7. In the present paper we extend the scope of Šik’s theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.:Modular Lattice of Locally Finite Length, Acta Sci. Math.37(1975), 79–82] cannot be strengthened is analogous way.

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].

Complemented Modular Lattices

1959 ◽  
Vol 11 ◽  
pp. 481-520 ◽  
Author(s):  
Ichiro Amemiya ◽  
Israel Halperin
Keyword(s):  
Modular Lattice ◽  
Modular Lattices ◽  
Von Neumann ◽  
Upper Continuous ◽  
Continuous Geometry

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

scholarly journals A Semimodular Imbedding of Lattices

1960 ◽  
Vol 12 ◽  
pp. 582-591 ◽  
Author(s):  
D. T. Finkbeiner
Keyword(s):  
Modular Lattice ◽  
Dimensional Lattice ◽  
General Lattice ◽  
Basic Properties ◽  
Arithmetic Properties ◽  
Finite Dimensional ◽  
Unpublished Work ◽  
Restricted Type ◽  
Point Lattice

The study of structural or arithmetic properties of a general lattice often can be facilitated by imbedding as a sublattice of a lattice of a more restricted type whose properties are known. However, if is too restricted, a general imbedding is impossible; for example, cannot be modular because , as a sublattice of , would then have to be modular. One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110). In the present paper Dilworth's imbedding process is modified to obtain a sharper result: Any finite dimensional lattice is isometrically isomorphic to a sublattice of a semi-modular lattice which has the same number of points as and which preserves basic properties of the join-irreducible arithmetic of .

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 111-114
Author(s):  
F. Okoh
Keyword(s):  
Rational Functions ◽  
Indecomposable Module ◽  
Torsion Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
The Status ◽  
Hereditary Algebras ◽  
Divisible Modules ◽  
Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

Modular C11 lattices and lattice preradicals

2017 ◽  
Vol 16 (06) ◽  
pp. 1750116 ◽  
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.

scholarly journals Two-sided localization in semiprime FBN rings

1984 ◽  
Vol 30 (2) ◽  
pp. 179-191
Author(s):  
M. H. Upham
Keyword(s):  
Torsion Theory ◽  
Prime Ideals ◽  
Closed Set ◽  
Left And Right ◽  
Torsion Theories ◽  
Quotient Rings

The main result of this paper is that the left- and right-quotient rings at a hereditary link closed set of prime ideals of a semiprime fully bounded Noetherian (FBN) ring coincide. This was a result already known for nonsemiprime FBN rigns, but a question left open in the semiprime case. A cornerstone of our approach is that the torsion theory determined by a link-closed hereditary set of prime ideals in an FBN ring is “nice”, but not necessarily perfect. Some conditions which do produce perfect torsion theories are investigated.

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