Mathematics

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Research from the Department of Mathematics. The department website may be found at https://www.ndsu.edu/math/

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Browsing Mathematics by Subject "Commutative rings."

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    Applications of Groups of Divisibility and a Generalization of Krull Dimension
    (North Dakota State University, 2011) Trentham, William Travis
    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.
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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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    Colorings of Zero-Divisor Graphs of Commutative Rings
    (North Dakota State University, 2015) Ramos, Rebecca Elizabeth
    We will focus on Beck’s conjecture that the chromatic number of a zero-divisor graph of a ring R is equal to the clique number of the ring R. We begin by calculating the chromatic number of the zero-divisor graphs for some finite rings and characterizing rings whose zero-divisor graphs have finite chromatic number, known as colorings. We will discuss some properties of colorings and elements called separating elements, which will allow us to determine that Beck’s conjecture holds for rings that are principal ideal rings and rings that are reduced. Then we will characterize the finite rings whose zero-divisor graphs have chromatic number less than or equal to four. In the general case, we will discuss a local ring that serves as a counterexample to Beck’s conjecture.