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Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.

The cardinality of the set of units, and of the set of equivalence classes of primes in non-trivial Euclidean domains is discussed with reference to the categories "finite" and "infinite." It is shown that no Euclidean domains exist for which both of these sets are finite. The other three combinations are possible and examples are given. For the more general Euclidean rings, the first combination is possible and examples are likewise given. Prime factorization is also discussed in both Euclidean rings and Euclidean domains. For Euclidean rings, an alternative definition of prime elements in terms of associates is compared and contrasted to the usual definitions.

by Leung Chi Kwan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-135). / Preface --- p.ii / Chapter 1 --- Introduction to Module Equivalence --- p.1 / Chapter 1.1 --- Introduction and Preliminaries --- p.1 / Chapter 2 --- Some Classical Results --- p.12 / Chapter 2.1 --- Morita Theorem --- p.12 / Chapter 2.2 --- Puller Theorem --- p.13 / Chapter 2.3 --- The Equivalence Mod-A ~Im(TP) --- p.29 / Chapter 2.4 --- The Equivalence Im(HP)~Im(Tp) --- p.33 / Chapter 3 --- *-modules and Tilting Modules --- p.39 / Chapter 3.1 --- The Equivalence Cogen(KA)~Gen(PR) --- p.39 / Chapter 3.2 --- Torsion Theories and *-modules --- p.56 / Chapter 3.3 --- The Structure of *-modules --- p.60 / Chapter 3.4 --- Characterizations of Tilting Modules --- p.65 / Chapter 4 --- Equivalences and Dualities --- p.85 / Chapter 4.1 --- The Equivalence PA~IR --- p.85 / Chapter 4.2 --- The Equivalence FGP-A ~FCI-R --- p.93 / Chapter 5 --- Torsion Theories Induced by Tilting Modules --- p.100 / Chapter 5.1 --- The Tilting Theorem --- p.100 / Chapter 5.2 --- Tilting Torsion Theories --- p.113 / Chapter 5.3 --- Isomorphisms of Endomorphism Rings of Tilting Modules --- p.122 / References --- p.133

Thesis (Ph. D.)--University of Idaho, 2006. / Abstract. "April 2006." Includes bibliographical references (leaf 69). Also available online in PDF format.

Thesis (Ph.D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 58-06, Section: B, page: 3073. Adviser: B. J. Mueller.

Thesis (M.S.)--University of Texas at El Paso, 2009. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2008. / Includes bibliographical references.