In this dissertation, we examine atomicity in rings with zero divisions. We begin by examining the relationship between a ring’s level of atomicity and the highest level of irreducibility shared by the ring’s irreducible elements. Later, we chose one of the higher forms of atomicity and identify ways of building large classes of examples of rings that rise to this level of atomicity but no higher. Characteristics of the various types of irreducible elements will also be examined. Next, we extend our view to include polynomial extensions of rings with zero divisors. In particular, we focus on properties of the three forms of maximal common divisors and how a ring’s classification as an MCD, SMCD, or VSMCD ring affects its atomicity. To conclude, we identify some unsolved problems relating to the topics discussed in this dissertation.
| Identifer | oai:union.ndltd.org:ndsu.edu/oai:library.ndsu.edu:10365/28905 |
| Date | January 2011 |
| Creators | Trentham, Stacy Michelle |
| 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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