published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

The applications of generalized inverses of matrices appear in many fields like applied mathematics, statistics and engineering [2]. In this thesis, we discuss generalized inverses of matrices over Ore polynomial rings (also called Ore matrices). We first introduce some necessary and sufficient conditions for the existence of {1}-, {1,2}-, {1,3}-, {1,4}- and MP-inverses of Ore matrices, and give some explicit formulas for these inverses. Using {1}-inverses of Ore matrices, we present the solutions of linear systems over Ore polynomial rings. Next, we extend Roth's Theorem 1 and generalized Roth's Theorem 1 to the Ore matrices case. Furthermore, we consider the extensions of all the involutions ψ on R(x), and construct some necessary and sufficient conditions for ψ to be an involution on R(x)[D;σ,δ]. Finally, we obtain two different explicit formulas for {1,3}- and {1,4}-inverses of Ore matrices. The Maple implementations of our main algorithms are presented in the Appendix. / May 2017

by Chan Suk Ha Iris. / Bibliography: leaves 84-88 / Thesis (M.Ph.)--Chinese University of Hong Kong

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 49). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Oregon, 2004. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 85-86). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1941. / Typescript. Includes abstract and vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf [61]).

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.

Let K be any field and Q be the rationals. Define K^1[X] = {f(X) e K[X]| the coefficient of X in f(X) is zero} and Q^1β[X] = {f(X) e Q[X]| the coefficent of β1(X) in the binomial expansion of f(X) is zero}, where {β1(X)}^∞ i=0 are the well-known binomial polynomials. In this work, I establish the following results: K^1[X] and Q^1β[X] are one-dimensional, Noetherian, non-Prüfer domains with the two-generator property on ideals. Using the unique factorization structure of the overrings K[X] and Q[X], the nonprincipal ideal structures of both rings are characterized, and from this characterization, necessary and sufficient conditions are found for a nonprincipal ideal to be invertible. The nonprincipal invertible ideals are then characterized in terms of the coefficients of the generators, and an explicit formula for the inverse of any proper invertible ideal is found. Finally, the class groups of both rings are shown to be torsion free abelian groups. Let n be any nonnegative integer. Results similar to the above are found in the generalizations of these two rings, K^n[X] and q^nβ[X], where the coefficients on the first n nonconstant basis elements are zero. For the domains K^1[X] and Q^1β[X], the property of strong two-generation is explored in detail and the following results are established: 1. K^1[X] and Q^1β[X] are not strongly two-generated, 2. In either ring, any polynomial with a constant term, or of degree two or three is a strong two-generator. 3. In K^1[X] any polynomial divisible by X^4 is not a strong two-generator, 4. An ideal I in K^1[X] or Q^1β[X] is strongly two-generated if and only if it is invertible.

Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors. In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.

Thesis (M.S.)--Missouri State University, 2008. / "August 2008." Includes bibliographical references (leaf 51). Also available online.