Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2009. / Submitted by Allan Wanick Motta (allan_wanick@hotmail.com) on 2010-05-17T18:58:10Z
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Previous issue date: 2009 / Seja F um corpo e seja A a F- algebra associativa livre (sem unidade) com
geradores livres x1; x2; :::. Seja f = f(x1; :::; xn) 2 A e seja G uma algebra
associtativa sobre F. Dizemos que f = 0 e uma identidade polinomial (ou
apenas uma identidade) em G se f(g1; :::; gn) = 0 para todos g1; :::; gn 2 G.
Dois sistemas de identidades polinomiais fui = 0 j i 2 Ig e fvj = 0 j j 2 Jg s~ao equivalentes se toda F- algebra associativa satisfazendo todas as identidades
ui = 0 satisfaz todas as identidades vj = 0 e vice-versa. Se o sistema
de identidades polinomiais fui = 0 j i 2 Ig e equivalente a algum sistema
nito de identidades, dizemos que o sistema fui = 0 j i 2 Ig tem base nita.
Nesta disserta c~ao, faremos um estudo detalhado de dois sistemas de
identidades polinomiais que n~ao possuem base nita, ou seja, que n~ao s~ao
equivalentes a um conjunto nito de identidades. O primeiro deles consiste
num sistema de identidades polinomiais que n~ao tem base nita em algebras
associativas (sobre um corpo de caracter stica 2) sem unidade e com unidade,
enquanto o segundo vale apenas para algebras associativas (sobre um corpo de
caracter stica 2) sem unidade e cont em a identidade x6 = 0. Esta disserta c~ao
foi baseada nos artigos [7] e [8] de Gupta e Krasilnikov, e no cap tulo 3 do
livro Free Algebras and PI-Algebras do Drensky [4]. ___________________________________________________________________________________________ ABSTRACT / Let F be a eld and let A be the free associative F-algebra (without
1) on free generators x1; x2; :::. Let f = f(x1; :::; xn) 2 A and let G be an
associative algebra over F. We say that f = 0 is a polynomial identity (or
an identity) in G if f(g1; :::; gn) = 0 for all g1; :::; gn 2 G. Two systems of
polynomial identities fui = 0 j i 2 Ig and fvj = 0 j j 2 Jg are equivalent
if every associative F-algebra satisfying all the identities ui = 0 satis es all
the identities vj = 0 and vice versa. If a system of polynomial identities
fui = 0 j i 2 Ig is equivalent to some nite system of identities, we say that
the system fui = 0 j i 2 Ig has a nite basis or is nitely based.
In this dissertation, we study in detail two systems of polynomial identites
that are not nitely based, that is, they are not equivalent to a nite
set of identities. The rst one consists of a system of polynomial identities
that has no nite basis in associative algebras (over a eld of characteristic
2) with or without unity, whereas the second one works only in non-unitary
associative algebras (over a eld of characteristic 2) and contains the identity
x6 = 0. This dissertation was based on the articles [7] and [8] by Gupta and
Krasilnikov, and the chapter 3 from the book Free Algebras and PI-Algebras
by Drensky [4].
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.unb.br:10482/4707 |
| Date | January 2009 |
| Creators | Resende, Marcos Mesquita |
| Contributors | Krassilnikov, Alexei |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis |
| Source | reponame:Repositório Institucional da UnB, instname:Universidade de Brasília, instacron:UNB |
| Rights | info:eu-repo/semantics/openAccess |
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