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Doctoral Thesis
DOI
10.11606/T.45.2006.tde-30012009-163028
Document
Author
Full name
Antonio Calixto de Souza Filho
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2006
Supervisor
Committee
Juriaans, Orlando Stanley (President)
Brumatti, Paulo Roberto
Cortes, Wagner de Oliveira
Milies, Francisco Cesar Polcino
Zalesski, Pavel
Title in Portuguese
Sobre uma classificação dos anéis de inteiros, dos semigrupos finitos e dos RA-loops com a propriedade hiperbólica
Keywords in Portuguese
álgebra de semigrupo
álgebra dos quatérnios
anel de grupo
grupo
grupo hiperbólico
inteiros algébricos
loop
ordens
semigrupo
teorema de estrutura
unidade
Abstract in Portuguese
Apresentamos duas construções para unidades de uma ordem em uma classe de álgebras de quatérnios que é anel de divisão: as unidades de Pell e as unidades de Gauss. Classificamos os anéis de inteiros de extensões quadráticas racionais, $R$, cujo grupo de unidades $\U (R G)$ é hiperbólico para um certo grupo $G$ fixado. Também classificamos os semigrupos finitos $S$, tal que, para a álgebra unitária $\Q S$ e para toda $\Z$-ordem $\Gamma$ de $\Q S$, o grupo de unidades $\U (\Gamma)$ é hiperbólico. Nesse mesmo contexto, classificamos os {\it RA}-loops $L$ cujo loop de unidades $\U (\Z L)$ não contém um subgrupo abeliano livre de posto dois.
Title in English
On a classification of the integral rings, finite semigroups and RA-loops with the hyperbolic property
Keywords in English
algebraic integers
group
group ring
hyperbolic group
loop
orders
quaternion algebra
semigroup
semigroup algebra
structure theorem
unit
Abstract in English
For a given division algebra of a quaternion algebra, we construct and define two types of units of its $\Z$-orders: Pell units and Gauss units. Also, for the quadratic imaginary extensions over the racionals and some fixed group $G$, we classify the algebraic integral rings for which the unit group ring is a hyperbolic group. We also classify the finite semigroups $S$, for which all integral orders $\Gamma$ of $\Q S$ have hyperbolic unit group $\U(\Gamma)$. We conclude with the classification of the $RA$-loops $L$ for which the unit loop of its integral loop ring does not contain a free abelian subgroup of rank two.
 
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Publishing Date
2009-03-20
 
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