Issued also as thesis, Bonn. / Added t.p. with thesis statement. Includes bibliographical references (p. 164-165).

Inaug.-Diss.--Bonn.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1954. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 43-44).

Thesis--University of Wisconsin-Madison. / Includes bibliographical references (p. 223-226).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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

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

Let C(X,E) denote the set of all continuous functions from a topological space X into a topological space E. R. Engelking and S. Mrowka [2] proved that for any E-completely regular space X [Definition 1.1], there exists a unique E-compactification [formula omitted] [Definitions 2.1 and 3.1] with the property that every function f in C(X,E) has an extension f in [formula omitted]. It is proved that if E is a (*)-topological division ring [Definition 5-5] and X is an E-completely regular space, then [formula omitted] is the same as the space of all E-homomorphisms [Definition 5.3] from C(X,E) into E. Also, we establish that if E is an H-topological ring [Definition 6.1] and X, Y are E-compact spaces [Definition 2.1], then X and Y are homeomorphic if, and only if, the rings C(X,E) and C(Y,E) are E-isomorphic [Definition 5.3]. Moreover, if t is an E-isomorphism from C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms from Y onto X with the property that [formula omitted] for all f in C(X,E), where π is the identity mapping on X and t is a certain mapping induced by t. In particular, the development of the theory of C(X,E) gives a unified treatment for the cases when E is the space of all real numbers or the space of all integers. Finally, for a topological ring E, the bounded subring C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs to C*(X,E) if for any O-neighborhood U in E, there exists a 0-neighborhood V in E such that f[X]•V c U and V•f[X] c U. The analogous results for C*(X,E) follow closely the theory of C(X,E); namely, for any E*-completely regular space X [Definition 9.5], there exists an E*-compactification [formula omitted] of X such that every function f in C (X,E) has an extension f in [formula omitted] when E is the space of all nationals, real numbers, complex numbers, or the real quaternions, [formula omitted] is just the space of all E-homomorphisms from C*(X,E) into E. This is also valid for a topological ring E which satisfies certain conditions. Also, two E*-compact spaces [Definition 10.1] X and Y are homeomorphic if, and only if, the rings C*(X,E) and C*(Y,E) are E-isomorphic, where E is any H*-topological ring [Definition 12.8]. / Science, Faculty of / Mathematics, Department of / Graduate

We are concerned with constructing examples of maximal abelian von Neumann subalgebras (MA subalgebras) in hyperfinite factors of type III. Our results will show that certain phenomena known to hold for the hyperfinite factor of type 11₁ also hold for type III factors. Let M and N be subalgebras of the factor α . We call M and N equivalent if M is the image of N by some automorphism of α . Let N(M) denote the subalgebra of α generated by all those unitary operators in α which induce automorphisms of M, and let N²(M), N³(M),... be defined in the obvious inductive fashion. Following J. Dixmier and S. Anastasio, we call a MA subalgebra M of α singular if N(M) = M, regular if N(M) = α, semi-regular if N(M) is a factor distinct from α, and m-semi-regular (m ≥ 2) if N(M),. . .N(m-1)(M) are not factors but N(m)(M) is a factor. The MA subalgebras of the hyperfinite 11₁ factor β have received much attention in the literature, in the papers of J. Dixmier, L. Pukanszky, Sister R. J. Tauer, and S. Anastasio. It is known that β contains a MA subalgebra of each type. Further, β contains pairwise inequivalent sequences of singular, semi-regular, 2-semi-regular, and 3-semi-regular MA subalgebras. The only hitherto known example of a MA subalgebra in a type III factor is regular. In 1956 Pukanszky gave a general method for constructing MA subalgebras in a class of (probably non-hyperfinite) type III factors. Because of an error in a calculation, the types of these subalgebras is not known. The main result of this thesis is the construction, in each of the uncountably many mutually non-isomorphic hyperfinite type III factors of R. Powers, of: (i) a semi-regular MA subalgebra (ii) two sequences of mutually inequivalent 2-semi-regular MA subalgebras 1 (iii) two sequences of mutually inequivalent 3-semi-regular MA subalgebras. Let α denote one of these type III factors and let β denote the hyperfinite 11₁ factor. Roughly speaking, whenever a non-singular MA subalgebra of β is constructed by means of group operator algebras, our method will produce a MA subalgebra of α of the same type. H. Araki and J. Woods have shown that α ⊗ β ≅ α, and it is therefore only necessary to construct MA subalgebras of α ⊗ β of the desired type. We obtain MA subalgebras of α ⊗ β by tensoring a MA subalgebra in α with one in β. In order to determine the type of such a MA subalgebra, we realize β as a constructible algebra and then regard α ⊗ β as a constructible algebra; this allows us to consider operators in α ⊗ β as functions from a group into an abelian von Neumann algebra. As a corollary to our calculations, we are able to construct mutually inequivalent sequences of 2-semi-regular and 3-semi-regular MA subalgebras of the hyperfinite 11₁ factor which differ from those of Anastasio. / Science, Faculty of / Mathematics, Department of / Graduate