This thesis carries out some of classical integration theory in the context of an operator algebra. The starting point is measure on the projections of an abelian von Neumann algebra. This yields an integral on the self-adjoint operators whose spectral projections lie in the algebra. For this integral a Radon-Nikodym theorem, as well as the usual convergence theorems is proved. The methods and results of this thesis generalize, to non-commutative von Neumann Algebras [2, 3, 5]. (1) J. Dixmier Les Algèbres d'Opérateurs dans l'Espace Hilbertien. Paris, 1957. (2) H.A. Dye The Radon-Nikodym theorem for finite rings of operators, Trans. Amer. Math. Soc, 72, 1952, 243-230. (3) F.J. Murray and J. von Neumann, On Rings of Operators, Ann. Math. 37, 1936, 116-229. (4) F. RIesz and B. v. Sz.-Nagy, Functional Analysis, New York, 1955. (5) I.E. Segal A non-commutative extension of abstract integration, Ann. of Math. (2) 57, 1953, 401-457. / Science, Faculty of / Mathematics, Department of / Graduate

We study the relations between algebraic properties of certain rings of functions and topological properties of the spaces on which the functions are defined. We begin by considering the relation between ideals of rings of functions and z-filters. Let [fomula omitted] be the ring of all m-times differentiable functions on a [formula omitted] differentiable n-manifold X , [formula omitted] the ring of all Lc-functions on a metric space X , and [formula omitted] the ring of all analytic functions on a subset X of the complex plane. It is proved that two m-(resp. Lc-) realcompact spaces X and Y are [formula omitted] diffeomorphic (resp. Lc-homeomorphic) iff [formula omitted] are ring isomorphic. Again if X and Y are m-(resp. Lc-) realcompact spaces, then X can be [formula omitted] (resp.Lc-) embedded as an open [resp. closed] subset in Y iff [formula omitted] homomorphic image of [formula omitted]. The subrings of [formula omitted] which determine the [formula omitted] diffeomorphism (resp. Lc-homeomorphism) of the spaces are studied. We also establish a representation for a transformation, more general than homomorphism, from a ring of [formula omitted] differentiable functions to another ring of [formula omitted] differentiable functions. Finally, we show that, for arbitrary subsets X and Y of the complex plane, if there is a ring isomorphism from [formula omitted] which is the identity on the constant functions, then X and Y are conformally equivalent. / Science, Faculty of / Mathematics, Department of / Graduate

M.Sc. (Mathematics) / Please refer to full text to view abstract

This paper is devoted to a discussion of certain aspects of an ideal theory of commutative rings. The material is divided into three chapters. Chapter I discusses necessary definitions and concepts. Chapter II deals with a theory of so called prime ideals and Chapter III is concerned with the theory of primary ideals. The entire theory is a generalization of the ideal theory of the ring of integers. / Advisor: N. Heerema, Professor Directing Paper. / Typescript. / "January, 1955." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 30).

Given a set A a (internal) law of composition on A is a function on AxA into A. A set endowed with one or more such laws of composition which in turn may be subject to certain axioms (e. g. associativity, commutativity) is termed an algebraic structure. The most fundamental algebraic structures are the group (which has a single law of composition) and the ring (which has two laws of composition). The structures of the "field" and "domain of integrity" are merely particular cases of the ring. This paper is devoted to an exposition of the basic algebraic theory of ring structure with some attention being devoted to the above cited particular cases of rings as well as to certain rather special substructures (ideals) of rings. In this development use is made throughout of the notion of the "algebraic image" of a structure, that is of homomorphism and isomorphism theory. This paper lays the groundwork for, but does not embark upon, the study of the structure theory of rings; which theory has been so beautifully developed in the last three decades by Artin, Noether and Jacobson. / Advisor: Robert N. Tompson, Professor Directing Paper. / Typescript. / "August, 1954." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 39).

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