Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

The ℵ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields

1978 ◽  
Vol 43 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields

AbstractLet n ≥ 3. The following theorems are proved.Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable.Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that K ⊨ φ if and only if R φ σ(φ).Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.

scholarly journals Maximal subalgebras of finite-dimensional algebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1283-1304 ◽  
Author(s):  
Miodrag Cristian Iovanov ◽  
Alexander Harris Sistko
Keyword(s):  
Associative Algebra ◽  
Complete Classification ◽  
Maximal Subalgebras ◽  
Incidence Algebras ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Closed Fields ◽  
Representations Of Algebras ◽  
Algebraically Closed Fields ◽  
Full Classification

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

scholarly journals Isomorphism of generalized triangular matrix-rings and recovery of tiles

2003 ◽  
Vol 2003 (9) ◽  
pp. 533-538 ◽  
Author(s):  
R. Khazal ◽  
S. Dăscălescu ◽  
L. Van Wyk
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Local Rings ◽  
Isomorphism Theorem ◽  
Matrix Rings ◽  
Recovery Result

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents0and1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

scholarly journals On the involution fixity of exceptional groups of Lie type

2018 ◽  
Vol 28 (03) ◽  
pp. 411-466 ◽  
Author(s):  
Timothy C. Burness ◽  
Adam R. Thomas
Keyword(s):  
Algebraic Geometry ◽  
Fixed Points ◽  
Algebraic Groups ◽  
Natural Analogue ◽  
Transitive Action ◽  
Exceptional Groups ◽  
Closed Fields ◽  
Suzuki Group ◽  
Algebraically Closed Fields ◽  
Groups Of Lie Type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

scholarly journals Representation-Directed Diamonds

2001 ◽  
Vol 4 ◽  
pp. 14-21
Author(s):  
Peter Dräxler
Keyword(s):  
Finite Dimensional Algebra ◽  
Program System ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Positive Roots ◽  
Algebraically Closed Fields

AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

scholarly journals Expansions of fields by angular functions

2008 ◽  
Vol 7 (4) ◽  
pp. 735-750 ◽  
Author(s):  
David M. Evans
Keyword(s):  
Model Theory ◽  
Angular Function ◽  
Existentially Closed ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractThe notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two ‘counterexamples’ from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

LINEAR RECURRING SEQUENCES OVER NONCOMMUTATIVE RINGS

2012 ◽  
Vol 11 (02) ◽  
pp. 1250040 ◽  
Author(s):  
AHMED CHERCHEM ◽  
TAREK GARICI ◽  
ABDELKADER NECER
Keyword(s):  
Recurrence Relation ◽  
Division Ring ◽  
Matrix Ring ◽  
Noncommutative Ring ◽  
Commutative Case ◽  
Noncommutative Rings ◽  
Linear Recurring Sequences ◽  
Finite Dimensional ◽  
The Stability

Contrary to the commutative case, the set of linear recurring sequences with values in a module over a noncommutative ring is no more a module for the usual operations. We show the stability of these operations when the ring is a matrix ring or a division ring. In the case of a finite dimensional division ring over its center, we give an algorithm for the determination of a recurrence relation for the sum of two linear recurring sequences.

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