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Subject: Integral domains.

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Now showing 1 - 2 of 2
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    Atomicity in Rings with Zero Divisors
    (North Dakota State University, 2011) Trentham, Stacy Michelle
    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.
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    Ramification and Infinite Extensions of Dedekind Domains
    (North Dakota State University, 2010) Hashbarger, Carl Stanley
    This dissertation presents methods for determining the behavior of prime ideals m an integral extension of a Dedekind domain. One tool used to determine this behavior is an algorithm that computes which prime ideals ramify in a finite separable extension. Other results about factorization of prime ideals are improved and applied to finite extensions. By considering a set of finite extensions whose union is an infinite extension, it is possible to predict ideal factorization in the infinite extension as well. Among other things, this ideal factorization determines whether a given infinite extension is almost Dedekind. These methods and results yield some interesting facts when they are demonstrated on a pair of classical rings of algebraic number theory.