On representability of *-regular rings and modular ortholattices [Elektronische Ressource] / von Florence Micol

Sujets

• Mathematics

Informations

• Publié par: technischen_universitat_darmstadt
• Publié le: 01 janvier 2003
• Nombre de lectures: 25
• Langue: English

Extrait

On representability
of-regular rings
and modular ortholattices
Vom Fachbereich Mathematik
der Technischen Universit at Darmstadt
zur Erlangung des Grades einer
Doktorin der Naturwissenschaften
(Dr. rer. nat)
genehmigte Dissertation
von
Dipl.-Math. Florence Micol
aus Saint-Etienne
Referent: Prof. Dr. Chr. Herrmann
Korreferent: Prof. Dr. K. H. Neeb
Tag der Einreichung: 9. Januar 2003
Tag der mundlic hen Prufung: 31. Januar 2003
Darmstadt 2003
D17In this thesis a proof is given that simple modular ortholattices possessing a
chain with at least ve elements (or four if they are arguesian) are
coordinatizable by a-regular ring also with respect to the orthocomplementation.
This is based on the fact that their lattice reduct possesses a large partial
three-frame and hence satis es a stricter condition of coordinatization
yielding the involution on the coordinatizing ring. Simple modular ortholattices
play an important role in the equational theory of modular ortholattices,
since any variety of modular ortholattices is generated by its simple
members, as shown by Herrmann and Roddy.
As a second main result, a characterization of the smallest classV
ofregular rings containing the classA of artinian-regular rings and closed
under homomorphic images (H), products (P ) and regular substructures
(S ) is set up. In fact, the elements ofV are exactly the-regular ringsr
that can be embedded into an atomic-regular ring, resp. the rings
that can be embedded into a product of rings of endomorphisms of some
vector spaces with scalar product such that the involution in the regular ring
corresponds to the adjunction of endomorphisms. Finally,V is obtained as
S HS PA.r r
Eines der Hauptergebnisse dieser Arbeit lautet: die einfachen modularen
Orthoverb ande, die eine Kette mit mindestens funf Elementen haben (oder vier,
falls sie arguesisch sind), sind koordinatisierbar durch einen-regul aren Ring
auch bezuglic h der Orthokomplementierung. Fur den Beweis wird benutzt,
dass das Redukt als Verband einen gro en partiellen 3-Rahmen hat, und
deshalb eine striktere Koordinatisierungsbedingung erfullt, die die Existenz der
Involution auf dem koordinatisierenden Ring garantiert. Die einfachen
modularen Orthoverb ande spielen tats achlich eine gro e Rolle in der
Gleichungstheorie der modularen Orthoverb ande, da nach Herrmann und Roddy jede
Variet at modularer Orthoverb ande von ihren einfachen Elementen erzeugt
wird.
Ein zweites Hauptergebnis ist die Charakterisierung der kleinsten
Klasse-regul arer Ringe V, die die KlasseA der artin’schen-regul aren Ringe umfasst,
und unter homomorphen Bildern (H), Produkten (P ) und regul aren
Unterstrukturen (S ) abgeschlossen ist. Es a tl sich zeigen, dass ein -regul arerr
Ring genau dann zuV geh ort, wenn er sich in einen atomaren aren
Ring einbettena t;l und das ist aquivalent dazu, dass er sich in ein Produkt
von Endomorphismenringen ub er Vektorr aumen mit Skalarprodukt einbetten
a t,l sodass die Involution auf dem Ring der Adjunktion fur die
Endomorphismen entspricht. Schlie lich erh alt man die Klasse V als S HS PA.r rContents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2 Notation and conventions . . . . . . . . . . . . . . . . . . . . 6
1 General framework and prerequisites 8
1.1 Relating-regular rings and MOLs . . . . . . . . . . . . . . . 8
1.1.1 Basic de nitions . . . . . . . . . . . . . . . . . . . . . . 8
1.1.2 The principal ideals of a-regular ring . . . . . . . . . 9
1.2 The class of-regular rings as a -variety . . . . . . . . . . . . 12
1.2.1 Relative varieties . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Application for-regular rings . . . . . . . . . . . . . . 12
1.2.3 Congruences and ideals . . . . . . . . . . . . . . . . . . 13
1.2.4 From R to eRe . . . . . . . . . . . . . . . . . . . . . . 14
2 Regular rings of contin. endomorphisms 15
2.1 Regular rings of . . . . . . . . . . . . . . . . . 15
2.1.1 Ideals of endomorphism rings and subspaces . . . . . . 15
2.1.2 Matrix representation . . . . . . . . . . . . . . . . . . . 16
2.2 Scalar products . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Vector spaces with scalar product . . . . . . . . . . . . 18
2.2.2 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Adjunction with respect to diagonalizable scalar products . . . 20
2.3.1 How to get diagonalizable scalar products . . . . . . . 20
2.3.2 Description of the ring of continuous endomorphisms . 22
2.4 Examples based on nitely generated projective modules . . . 23
2.4.1 Full rings of nite matrices over a -regular ring . . . . 23
2.4.2 An example for a generalized system of nn matrix
units over a regular ring . . . . . . . . . . . . . . . . . 24
2.5 Examples derived from at least countably based vector spaces 25
2.5.1 Sums of rings . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.2 The endomorphisms ( k) . . . . . . . . . . . . . . . . 26
2.5.3 Examples of representable-regular rings . . . . . . . . 27
2.6 The von Neumann example (continuous geometry) . . . . . . . 29
2CONTENTS 3
2.6.1 The index set . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.2 The vector space and the scalar product . . . . . . . . 30
2.6.3 The matrix ring . . . . . . . . . . . . . . . . . . . . . . 30
3 Representation of-regular rings 32
3.1 Atomic-regular rings and representability . . . . . . . . . . . 32
3.1.1 Atomic-regular rings . . . . . . . . . . . . . . . . . . 32
3.1.2 Representability . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 A more general notion of representability . . . . . . . . 35
3.1.4 Proof for Theorem 3.8 . . . . . . . . . . . . . . . . . . 36
3.1.5 Atomic extensions . . . . . . . . . . . . . . . . . . . . 42
3.2 Approximation by rings of matrices of nite size . . . . . . . . 43
3.2.1 A notion of approximation/convergence . . . . . . . . . 43
3.2.2 Approximation for rings of small matrices . . . . . . . 45
3.2.3 The class of sr-artinian rings . . . . . . . . . . . . . . . 47
4 Representation of MOLs 48
4.1 Varieties of MOLs . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 The variety of proatomic MOLs . . . . . . . . . . . . . 50
4.2 Coordinatization for MOLs . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Inducing lattice-isomorphisms . . . . . . . . . . . . . . 52
4.2.2 Strictly uniquely coordinatizable lattices . . . . . . . . 54
4.3 Consequences of Theorem 4.14 for varieties of MOLs . . . . . 56
4.4 Examples and counter-examples . . . . . . . . . . . . . . . . . 57
4.4.1 Non-uniquely coordinatizable MOLs . . . . . . . . . . 57
4.4.2 Some examples for non-coordinatizable MOLs . . . . . 58
4.4.3 The J onsson example (simple MOL without global
nframe ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Open questions 61
A 62
A.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.2 Results of J onsson . . . . . . . . . . . . . . . . . . . . . . . . 620.1 Introduction
Regular rings are rings that satisfy the axiom8x:9y:xyx =x.
They have been introduced by von Neumann in the late 1930s. For a
regular ringR, call the set of all principal right idealsL(R). This
happens to be a complemented modular lattice (CML). The CMLs that
are isomorphic to someL(R) are called coordinatizable. This notion
of coordinatization is closely related to the classical representation of
projective geometries as lattices of subspaces of a vector space: for a
regular ring R and an integer n, there is a natural isomorphism
bentween the lattice of submodules of the R-module R and the lattice of
principal ideals of the ring of nn-matrices over R.
A-regular ring is a regular ring equipped with an involution that
satis es the condition xx = 0 =) x = 0. The lattice of principal
right ideals then becomes a modular ortholattice (MOL). In the case
of projective geometries of nite dimension, this implies the existence
of a scalar product on the vector space.
In fact, it is possible to generalize the classical coordinatization
theorem of projective geometry to certain CMLs without atoms (atoms
correspond to points in projective geometries). First, von Neumann
has proved that a CML is (uniquely) coordinatizable by a regular ring
provided the lattice has a ‘global n-frame’. An analogous result holds
for MOLs with a-regular ring. The rst theorem was improved by
J onsson in the 1960s. J onsson weakened the condition of existence of
‘global n-frames’ to ‘large partial n-frames’, and he was able to show
that this condition holds in simple CMLs of higher dimension.
His ideas are used here to solve the problem of-coordinatization for
simple MOLs of higher dimension. In fact, simple MOLs of
dimension at least four or Arguesian MOLs of dimension at least three are
(strictly uniquely)-coordinatizable. Indeed the simple MOLs play an
important role in the equational theory of MOLs, since any variety of
MOLs is generated by its simple members, as shown by Herrmann and
Roddy.
On the other hand,-regular rings are interesting structures
on their own. For example, the involution may be the adjunction in
rings of endomorphisms over a vector space with respect to some