Groups of divisibility have played an important role in commutative algebra for many years. In 1932 Wolfgang Krull showed in [12] that every linearly ordered Abelian group can be realized as the group of divisibility of a valuation domain. Since then it has also been proven that every lattice-ordered Abelian group can be recognized as the group of divisibility of a Bezont domain. Knowing these two facts allows us to use groups of divisibility to find examples of rings with highly exotic properties. For instance, we use them here to find examples of rings which admit elements that factor uniquely as the product of uncountably many primes. In addition to allowing us to create examples, groups of divisibility can he used to characterize some of the most important rings most commonly encountered in factorization theory, including valuation domains, UFD's, GCD domains, and antimatter domains. We present some of these characterizations here in addition to using them to create many examples of our own, including examples of rings which admit chains of prime ideals in which there are uncountably many primes in the chain. Moreover, we use groups of divisibility to prove that every fragmented domain must have infinite Krull dimension.

Made available in DSpace on 2015-05-15T11:46:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 885143 bytes, checksum: 948ed501e70f201fbd41ae588572c5bc (MD5) Previous issue date: 2010-08-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We present in this work a complete classification of the sublattices of (Zn,+, ≥) which are not groups of divisibility. Thus we provide a new class of ordered filtered groups of which are not groups of divisibility. The sublattices presented here generalize the exemples of P.Jaffard and G. G. Bastos / Apresentamos nesse trabalho uma classificação completa de sub-reticulados de (Zn,+, ≥) que não são grupos de divisibilidade. Deste modo, nós fornecemos uma nova classe de grupos ordenados que são filtrados, mas não são grupos de divisibilidade. Os sub-reticulados aqui apresentados generaliza os exemplos de P. Jaffard e G. G. Bastos.