The primary objective of this thesis is to present a unified account of the various generalizations of the concept of ring of quotients given by K, Asano (1949), R. E. Johnson (1951), Y. Utumi (1956), G. D. Findlay and J. Lambek (1958). A secondary objective is to investigate how far the commutative localization can be carried over to the noncommutative case. We begin with a formulation of the notion of D-system of right ideals of a ring R. The investigation of the D-systems was motivated by the fact that each maximal right quotient ring of R consists precisely of semi R-homomorphisms into R with domains in a specific D-system of right ideals of R or of R¹, the ring obtained from R by adjoining identity. A nonempty family X of right ideals of R is called a D-system provided the following three conditions holds: D1. Every right ideal of R containing some member of X is in X. D2. For any two right ideals A and B of R belonging to X, φ⁻¹B belongs to X for each R-homomorphism φ of A into R. D3. If A belongs to X and if for each a in A there exists Ba in X, then the ideal sum of aBa (a in A) is in X, Each D-system X of right ideals of R induces a modular closure operation on the lattice L(M) of all submodules of an R-module M and hence gives rise to a set Lx(M) of closed submodules of M. We are able to set up an isomorphism between the lattice of all modular closure operations on L(R) and the lattice of all D-systems of right ideals of R and characterize the D-systems X used in Asano's, Johnson's and Uturai's constructions of«quotient rings in terms of properties of Lx(R). In view of the intimate relation between the rings of quotients of a ring R and the extensions of R-modules, we generalize the concepts of infective R-module, rational and essential extensions of an R-module corresponding to a D-system T of right ideals of R¹. The existence and uniqueness of the maximal Y-essential extension, minimal Y-injective extension and maximal Y-rational extension of an R-module and their mutual relations are established. Finally, we come to the actual constructions of various extensions of rings and modules. The discussions center around the centralizer of a ring over a module, the maximal essential and rational extensions and the different types of rings of right quotients. We include here also a partial, though not satisfactory, solution of the noncommutative localization problem. / Science, Faculty of / Mathematics, Department of / Graduate

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Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions. / by Mario V. Marshall. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 92-93). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Sept. 18, 2008). PDF text: v, 109 p. : ill. ; 459 K. UMI publication number: AAT 3303652. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. Description based on print version record. Includes bibliographical references.

Thesis (Ph.D.)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 65-69)

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.