Hypermatrices are defined. Elementary operations and properties are defined and discussed. A 4-ary Multiplication is defined for hypermatrices, consisting of multilinear mappings from ordered 4-tuples of hypermatrices to hypermatrices. This multiplication is the only such mapping satisfying two basic properties which we should like such an operation to have. Various properties and characterizations of Multiplication are discussed. Equivalence Classes of hypermatrices are defined and discussed. Starting with equivalence classes of a general nature, we are led to the definition of various types of Hyperdeterminants, themselves considered as being equivalence classes of hypermatrices. Operators and operations on hypermatrices are extended to hyperdeterminants. A generalization of the Cauchy-Binet Theorem for matrices is seen to hold for hypermatrices and their associated hyperdeterminants. Special systems of hypermatrices are seen to constitute generalizations of the Complex and Quaternion Algebras, and some properties of these are discussed. / Science, Faculty of / Mathematics, Department of / Graduate

Several aspects of the theory of radical classes in associative ring theory are investigated. In Chapter three, the Andrunakievic-Rjabuhin construction of radicals by means of annihilators of modules is employed to define several radical properties. One of these is shown to be the "weak radical" of Koh and Mewborn. The relations between these radicals, their properties and some of their applications to the study of classical quotient rings are investigated. In Chapter four, the ideals of a ring K of the form R(K), for a hereditary radical, R, are studied. A closure operation on the lattice of ideals is introduced, and the "closed" ideals are precisely the ideals of this type. It is proved that the ascending and descending chain conditions on the closed ideals of a ring imply that the ring has only a finite number of closed ideals. In Chapter five, finite subdirect sums of rings are studied. The properties of hereditary radicals and of the various structure spaces, in a situation where one has a finite subdirect sum of rings, are investigated. / Science, Faculty of / Mathematics, Department of / Graduate

For any cardinal number K ≥2 and any non-empty class of rings ℛ we make the following definitions. The class ℛ(K) is the class of all rings R such that every subring of R which is generated by a set of cardinality strictly less than K is in ℛ . The class ℛg(K) is the class of all rings R such that every non-zero homomorphic image of R contains a non-zero subring in ℛ which is generated by a set of cardinality strictly less than K . Several properties of the classes ℛg(K) and ℛ(K) are determined. In particular, conditions are specified which imply that ℛ(K) is a radical class or a semi-simple class. Necessary and sufficient conditions that the class ℑ of all ℛg(K) semi-simple rings be equal to ℑ(K) are given. The classes ℛ(K) and ℛg(K) when K = 2 or K = (formula omitted)₀ are considered in detail for various classesℛ (including the cases when ℛ is one of the well-known radical classes). In all cases when ℛ is one of the well-known radical classes it Is shown that ℛ(2) and ℛ(formula omitted) are radical classes and whenever they contain all nilpotent rings they are shown to be special radical classes. Those radical classes ℛ(2) which are contained in FC (R € FC if and only if for all x € R , x is torsion) are characterized. Let ℛ be any radical class. The largest radical class (formula omitted) (if one exists) such that (formula omitted)(R) Ո ℛ(R) = (0) for all rings R is defined to be the local complement of ℛ̅̅ and is denoted by ℛ. If ℛ = ℛ(formula omitted) then the local complement ℛ exists and ℛ= ℛ(2) . The local complements of all radicals discussed are determined. We are able to apply some of these results in order to classify those classes of rings which are both semi-simple and radical classes. / Science, Faculty of / Mathematics, Department of / Graduate

The study of certain series of groups has greatly aided the development and understanding of group theory. Normal series and central series are particularly important. This paper attempts to define analogous concepts in the theory of rings and to study what interrelationships exist between them. Baer and Freidman have already studied chain ideals, the ring theory equivalent of accessible subgroups. Also, Kegel has studied weakly nilpotent rings, the ring theory equivalent of groups possessing upper central series. Some of the more important results of these authors are given in the first three sections of this paper. Power nilpotent rings, the ring theory equivalent of groups possessing lower central series, are defined in section 4. The class of power nilpotent rings is not homomorphically closed. However, it does possess many of the other properties that the class of weakly nilpotent rings has. In section 5 meta* ideal and U*-ring are defined in terms of descending chains of subrings of the given ring. Not every power nilpotent ring is a U*-ring. This is contrary to the result for semigroups. It is also shown that an intersection of meta* ideals is always a meta* ideal. It follows that not every meta* ideal is a meta ideal since the intersection of meta ideals is not always a meta ideal. Section 6 is concerned with rings in which only certain kinds of multiplicative decomposition take place. The rings studied here are called prime products rings and it is proved that all weakly nilpotent and power nilpotent rings are prime products rings. A result given in the section on U-rings suggests that all U-rings may be prime products rings. The class of prime products rings is very large but does not include any rings with a non-zero idempotent. The last section studies ring types which are defined analogously to group types. The study of which ring types actually occur is nearly completed here. Finally, it is shown that every weakly nilpotent ring has a ring type similar to that of some ring which is power nilpotent. This suggests (but does not prove) the conjecture that all weakly nilpotent rings are power nilpotent. / Science, Faculty of / Mathematics, Department of / Graduate

An algebraic system which satisfies all the ring axioms with the possible exceptions of commutativity of addition and the right distributive law is called a near-ring. This thesis is intended as a survey of radicals in near-rings, and an organization of the theory which has been developed to date. Because of the absence of the right distributive law, the zero element of a near-ring need not annihilate the near-ring from the left. If we impose the condition that 0 • p = 0 for all elements p of a near-ring P, then we call P a C-ring. This condition is ensured if we demand that the near-ring P be generated, as an additive group, by a set S of elements of P such that (P₁+ P₂)s = P₁s + P₂S for all P₁, P₂ in P, and s in S. In this case, P is said to be distributively generated by S. The work is divided into three main sections; the first deals with general near-rings, the second with C-rings, and the third with distributively generated near-rings. Appendix I gives a proof of a vital result for distributively generated near-rings, due to Laxton [11]; appendix II introduces a little used radical due to Deskins [6]; appendix III is included as a concrete example of a near-ring and its theory, due to Berman and Silverman [2]. / Science, Faculty of / Mathematics, Department of / Graduate

Since Kaplansky's first paper on the subject of P.I. rings appeared in 1948, many fruitful results have arisen from the study of such rings. This thesis attempts to present the most important of these results in a unified theory. Chapter I gives the basic notation, definitions, a number of small lemmas together with Kaplansky's incisive result on primitive P.I. rings. We investigate also the Kurosh problem for P.I. rings, providing for such rings an affirmative answer. A rather nice universal property for P.I. rings which ensures that all P.I. rings satisfy some power of the standard identity is proved. Chapter II deals with particular types of rings such as rings without zero divisors and prime rings and culminates in a pair of pretty results due to Posner and Procesi. We show that prime P.I. rings have a rather tight structure theory and in fact the restrictions on the underlying set of coefficients can in this case be relaxed to a very great extent. Chapter III is exclusively devoted to P.I. rings with involution. Although such rings are rather specialized much has been accomplished in this direction in recent years and many beautiful theorems and proofs have been established, especially by Amitsur and Martindale. The source material for chapter IV is primarily Procesi and Amitsur's work on Jacobson rings and Hilbert algebras. Application to Hilbert's Nullstellensatz and to the Burnside problem are considered. Finally, chapter V concerns itself completely with generalizations of the preceding four chapters. For the most part these results do not generalize entirely, but by reducing our demand on polynomial identities slightly, many remarkably fine results have been proved. / Science, Faculty of / Mathematics, Department of / Graduate

Boolean valued structures are defined and some of their properties are studied. Completeness and compactness theorems are proved and Lowenheim-Skolem theorems are looked at. It is seen that for any consistent theory T and cardinal number KT there is a model N of T a "universal" model) such that for any model M of T with M <, K, M can be written as a quotient of N. A theory T is shown to be open if and only if given structures M c N, if N is a model of T, then M is a model of T, T is shown to be existential if and only if the union of every chain of models of T is a model of T. The prefix problem and obstructions to elementary extensions are examined. Various forms of completeness are compared and, finally, an example is given where Boolean valued models are used to prove a theorem of Mathematics (Hilbert's 17-th Problem) without using the Axiom of Choice. Throughout, it is seen that good Boolean valued structures (for all Φ [(Ev[sub j])Φ][sub M] = I(Φ,a][sub M] for some a Є U[sub M]) behave very much like relational structures and much of the theory depends upon their existence. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis obtains information about Boolean algebras by means of the radical concept. One group of results revolves about the concept, theorems, and constructions of general radical theory. We obtain some subdirect product representations by methods suggested by the theory. A large number of specific radicals are defined, and their properties and inter-relationships are examined. This provides a natural frame-work for results describing what epimorphs an algebra can have. Some new results of this nature are obtained in the process. Finally, a contribution is made to the structure theory of complete Boolean algebras. Product decomposition theorems are obtained, some of which make use of chains of radical classes. / Science, Faculty of / Mathematics, Department of / Graduate

The integral cohomology algebra functor, H*( ;Z), was developed as an aid in distinguishing homotopy types. We consider the problem of when there are only a finite number of homotopy equivalence classes in the collection of simply connected, finite CW complexes, for which the cohomology algebra is isomorphic to a given algebra. We prove that, if we restrict ourselves to formal homotopy types, the set of such homotopy types is always finite. This is shown by using the concept of distance between homotopy types. We build model spaces so that the distance from a CW complex, whose cohomology is isomorphic to the given algebra, to one of the model spaces can be bounded. General results about distance then imply that the set of homotopy types is finite. Formality is a property of rational homotopy type and we use information obtained from calculations with the minimal models of Sullivan as a guide in the construction of spaces and maps. As a partial converse, we show that, for every nonformal space, X, there is a homology section, X', of X such that there are an infinite number of different homotopy types with cohomology algebras isomorphic to H*(X';Z) and with the same rational homotopy type as X'. The dual problem, in the sense of Eckmann-Hilton, is also shown to have similar answers. For the dual problem one would replace "cohomology algebra" with "Samuelson algebra", "formal" with "coformal", and "homology section" with "Postnikov section". The result applies to many naturally occuring spaces, such as topological groups, H-spaces, complex and quaternionic projective spaces, Kahler manifolds and MoiSezon spaces. / Science, Faculty of / Mathematics, Department of / Graduate

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