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.
| Identifer | oai:union.ndltd.org:ndsu.edu/oai:library.ndsu.edu:10365/28904 |
| Date | January 2011 |
| Creators | Trentham, William Travis |
| Publisher | North Dakota State University |
| Source Sets | North Dakota State University |
| Detected Language | English |
| Type | text/dissertation |
| Format | application/pdf |
| Rights | NDSU Policy 190.6.2, https://www.ndsu.edu/fileadmin/policy/190.pdf |
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