SOME ASPECTS OF MODULAR LATTICES WITH DUAL KRULL DIMENSION

2005 ◽  
Vol 04 (03) ◽  
pp. 237-244
Author(s):  
MARK L. TEPLY ◽  
SEOG HOON RIM
Keyword(s):  
Endomorphism Ring ◽  
Modular Lattice ◽  
Type Theorem ◽  
Krull Dimension ◽  
Local Domain ◽  
Module Theory ◽  
Modular Lattices ◽  
Atomic Lattices

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules.

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Unpublished Manuscript ◽  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 Math.37(1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving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.

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.

On the Word Problem for Orthocomplemented Modular Lattices

1989 ◽  
Vol 41 (6) ◽  
pp. 961-1004 ◽  
Author(s):  
Michael S. Roddy
Keyword(s):  
Word Problem ◽  
Division Ring ◽  
Modular Lattice ◽  
Modular Lattices ◽  
Commutative Field ◽  
Free Modular Lattice ◽  
Finitely Presented ◽  
Unsolvable Word Problem

In [16] Freese showed that the word problem for the free modular lattice on 5 generators is unsolvable. His proof makes essential use of Mclntyre's construction of a finitely presented field with unsolvable word problem [30]. (We follow Cohn [7] in calling what is commonly called a division ring a field, and what is commonly called a field a commutative field.) In this paper we will use similar ideas to obtain unsolvability results for varieties of modular ortholattices. The material in this paper is fairly wide ranging, the following are recommended as reference texts.

On rings whose nonunits are a unit multiple of a nilpotent

2016 ◽  
Vol 16 (07) ◽  
pp. 1750140
Author(s):  
Peter Vámos
Keyword(s):  
Vector Space ◽  
Finite Length ◽  
Endomorphism Ring ◽  
Polynomial Identity ◽  
Morita Equivalence ◽  
Complete Characterization ◽  
Krull Dimension ◽  
Endomorphism Rings ◽  
Matrix Rings ◽  
Skew Field

The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.

scholarly journals Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Marcin Łazarz
Keyword(s):  
Modular Lattice ◽  
Continuous Lattice ◽  
Atomic Lattice ◽  
Discrete Lattices ◽  
Upper Continuous Lattice ◽  
Upper Continuous ◽  
Strongly Atomic Lattice ◽  
Atomic Lattices

In the paper we investigate Birkhoff’s conditions (Bi) and (Bi*). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi*)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7 (the lattice S7). As a corollary we obtain a well known result of J. Jakub´ık from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S7 ≪ L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi*) if and only if L is a 4-cell lattice such that S7 ≪/ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  

scholarly journals Graded Modules as a Clean Comodule

2020 ◽  
Vol 12 (6) ◽  
pp. 66
Author(s):  
Nikken Prima Puspita ◽  
Indah Emilia Wijayanti ◽  
Budi Surodjo
Keyword(s):  
Group Ring ◽  
Endomorphism Ring ◽  
Module Theory ◽  
Graded Modules ◽  
Clean Ring ◽  
Graded Module

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi&acute;nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

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.

Hjelmslev Planes Derived from Modular Lattices

1969 ◽  
Vol 21 ◽  
pp. 76-83 ◽  
Author(s):  
Benno Artmann
Keyword(s):  
Modular Lattice ◽  
Hjelmslev Plane ◽  
Modular Lattices ◽  
Natural Embedding ◽  
Valuation Rings ◽  
A Chain ◽  
Homogeneous Basis

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.

scholarly journals ON COMPLETE CONGRUENCE LATTICES OF COMPLETE MODULAR LATTICES

1991 ◽  
Vol 01 (02) ◽  
pp. 147-160 ◽  
Author(s):  
R. FREESE ◽  
G. GRÄTZE ◽  
E. T. SCHMIDT
Keyword(s):  
Complete Lattice ◽  
Modular Lattice ◽  
Modular Lattices ◽  
Congruence Lattices ◽  
Congruence Relations

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In 1988, the second author announced the converse: every complete lattice L can be represented as the lattice of complete congruence relations of some complete lattice K. In this paper we improve this result by showing that K can be chosen to be a complete modular lattice.

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