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Almost Dedekind Domains and Atomicity
(North Dakota State University, 2012)
The objective of this dissertation was to determine the class of domains that are both almost Dedekind and atomic. To investigate this question we constructed a global object called the norm, and used it to determine ...
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015)
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
Gorenstein Dimension over Some Rings of the Form R [0 plus]C
(North Dakota State University, 2015)
Commutative algebra is the study of commutative rings and other abstract structures based on commutative rings. Modules can be viewed as a common generalization of several of those structures, and some invariants, e.g. ...
The Half-Factorial Property in Polynomial Rings
(North Dakota State University, 2014)
This dissertation investigates the following question: If R is a half-factorial domain (HFD) and x is an indeterminate, under what conditions is the polynomial ring R[x] an HFD? The question has been answered in a few ...
DG Homological Algebra, Properties of Ring Homomorphisms, and the Generalized Auslander-Reiten Conjecture
(North Dakota State University, 2013)
This dissertation contains three aspects of my research that are listed as three joint papers with my advisor and a solo paper in the bibliography [56]--[59]. Each paper will be discussed in a different chapter. Chapter 1 ...
Codualizing Modules and Complexes
(North Dakota State University, 2013)
See dissertation for full abstract and properly formatted mathematical formulas.
Variational Methods for Polycrystal Plasticity and Related Topics in Partial Differential Equations
(North Dakota State University, 2013)
In the first part of the thesis the effective yield set of ionic polycrystals is characterized by means of variational principles in L∞ that are associated to supremal functionals acting on matrix-valued divergence-free ...
Semidualizing DG Modules over Tensor Products
(North Dakota State University, 2015)
In this dissertation, we study rings: sets with addition, subtraction, and multiplication. One way to study a ring is by studying its modules: the algebraic objects the ring acts on. Since it is impractical to study all ...
Atomicity in Rings with Zero Divisors
(North Dakota State University, 2011)
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 ...
Ideal Graphs
(North Dakota State University, 2014)
In this dissertation, we explore various types of graphs that can be associated to a commutative ring with identity. In particular, if R is a commutative ring with identity, we consider a number of graphs with the vertex ...