The theory of nearrings has arisen in a variety of ways. There is a natural desire to generalise the theory of rings and skew fields by relaxing some of their defining axioms. It has also been the hope of some mathematicians that certain problems in group theory, particularly involving permutation groups and group representations, may perhaps be clarified by developing a coherent algebraic theory of nearrings. Moreover, there is an increasing recognition by mathematicians in many branches of the subject, both pure and applied, of the ubiquity of nearring like objects. The first steps in the subject were taken by Dickson and Zassenhans with their studies of 'nearfields', and by Wielandt with his classification of an important class of abstract nearrings. Papers by Frohlich, Blackett, Betsch and Laxton developed the theory considerably. Lately authors such as Beidleman, Ramakotaiah, Tharmanatram, Maxson, Malone and Clay have all added to our knowledge. The history of the subject has been strongly influenced by our knowledge of ring theory, and although this has often been beneficial it must not be overlooked that a number of important problems in nearring theory have no real parallel in the theory of rings. It is probably best to try to preserve a balance, and not to endeavour exclusively, either to generalise theorems from ring theory irrespective of their usefulness, or to ignore the theory of rings and attempt to formulate a completely independent theory. In many cases our results are generalisations of theorems from ringtheory but at certain important junctures we will explicitly use the fact that we are dealing with a nearring which is not a ring. This is a very interesting development in the subject. We proceed, in the first chapter, with a review of the terms and notation that will be used in this thesis. Where definitions and concepts are of a specialized or technical nature and only used in one section, it seems more sensible to postpone introducing them until a more natural point in the proceedings. Chapter 2 gives a summary of the results on the various radicals corresponding to the Jacobson radical for associative rings. Most of these results are well known and readily available in the literature. We also consider nearrings with one, or more, of these radicals zero. We defined, in Chapter 1, three different types of primitive nearring, which are all genuine generalisations of the ring theoretic concept. Of these three, the two most important are 2primitive and 0primitive nearrings. In Chapter 3, we examine 2primitive nearrings with certain natural conditions imposed on them. A theorem is obtained which could be considered to be the equivalent result for nearrings of the theorem classifying simple, artinian rings, due originally to Wedderburn and redeveloped by Jacobson. Chapters 4 and 5 deal with 0primitive nearrings satisfying certain conditions. Chapter 5 is a generalisation of Chapter 4, but we felt that the mathematical techniques involved would be clearer if the special case in Chapter 4 was expounded first. In these two chapters we classify a sizeable class of 0primitive nearrings with identity. and descending chain condition on right ideals. Several types of prime nearrings have been developed in the literature. In Chapter 6 we examine these and related concepts. In the theory of rings, Goldies' classification of prime and semiprime ring with ascending chain conditions, has been of immense importance. Whether such a result could be obtained in the theory of nearrings is a matter for conjecture, at the moment. We have made a start on the problem with the construction of a class of nearrings which behave in a very similar way to Prime rings with the Goldie chain conditions. This is the content of Chapter 7. The inspiration for its came mainly from the proof of Goldies' first theorem, due to C. Procesi, which is featured in Jacobson's book. (Jacobson [1]). Chapter 8, is an attempt to initiate the development of a theory of vector groups and nearalgebras which would play an important rölein the future theory of nearrings, in a way, perhaps, similar to the Ale vector spaces and algebras play in ring theory. This may lead, in time, to results on 2primitive nearrings with identity and a minimal right ideal, for example, or a Galois theory for certain 2primitive nearrings. For the former problem, the experience of the semigroup theorists (Hoehake [1] etc. ) may prove useful. Finally a note on the numbering of results and definitions etc. If a reference is made, containing only two numbers, e. g. 1.12 then this means, "item 12 of section 1 of the present chapter". If a reference reads: 3.1.12, then this means "item 12 of section 1 of Chapter 3.
