Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaf 67).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1976. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Bibliography: leaves 384-389.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M. Ph.)--University of Wisconsin--Madison, 1946. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.A.)--Boston University / A Dedekind ring R is defined as an integral domain which has the following three properties A. R is a Noetherian ring. B. Every prime ideal in R is a maximal ideal. C. R is integrally closed in its quotient field P. A Noetherian ring is a commutative ring for which the divisor chain condition is valid. If the divisor chain condition is valid in a ring r then every set of ideals A1 in R such that A1 C AI+1 properly, is a finite set. For Noetherian rings a theory of ideals may be developed. The main results of this theory are listed in the appendix [TRUNCATED]

M.Sc. (Mathematics) / Die eienskappe van 'n ring het 'n bepalende invloed op die eienskappe van die module oor daardie ring. So kan belangrike klasse van ringe gekarakteriseer word met behulp van module oar sodanige ringe. Origens blyk dat eienskappe van R-module wat in die algemeen nie saamval nie. weI saamval as ons R beperk tot sekere klasse van ringe. In die literatuur word die verwantskappe tussen 'n moduul en sy ring van skalare wyd verspreid aangetref. Die oogmerke van hierdie skripsie is om die belangrikste resultate wat die verwantskap tussen ringe uit 'n gegewe klas met hulle module aandui in een bron saam te bring vir sekere belangrike klasse van ringe. 'n Literatuurlys word vir verdere insae verskaf. Geen aanspraak word op volledigheid gemaak nie. Definisies word deurgaans gegee ter wille van volledigheid en ter wille van terminologie. Waar stellings en lemmas egter direk uit definisies volg. word geen bewys gegee nie. Bewyse word weI gegee in gevalle waar 'n besondere strategie. helderheid of bondigheid dit regverdig.

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Prior to 1990, the only well known ideal-hereditary Kurosh-Amitsur radicals in the variety of zero-symmetric near-rings were the Jacobson type radicals Iv(N) , where ∨∈{2,3} and the Brown-McCoy radical. In 1990, Booth, Groenewald and Veldsman introduced the concept of an equiprime near-ring which leads to an ideal-hereditary Kurosh-Amitsur radical in N∘. The concept of an equiprime near-ring generalizes the concept of a prime ring to near-rings. Although the search for more ideal-hereditary radicals of near-rings was apparently the original motivation for the introduction of equiprime near-rings, it became clear that these near-rings are interesting in their own right. It is our aim in this treatise to give an exposition of the many interesting properties of equiprime near-rings. We begin with a brief reminder of near-ring rudiments; giving basic definitions and elementary results which are necessary for understanding and development of subsequent chapters. With the basics out of the way, our main task begins with a consideration of equiprime, strongly and completely equiprime left ideals. It is noted that any zero-symmetric near-ring can be embedded in an equiprime near-ring. Moreover, the class of equiprime near-rings is shown to be hereditary. Open questions arising out of the study of equiprime near-rings are highlighted along the way. In Chapter 3 we consider well known examples of near-rings and determine when such near-rings are equiprime. This provides more insight into the nature of equiprime near-rings and is a fertile ground for the birth of examples and counterexamples which may be used to close or solve some open question within the literature. We also prove some results which generalize some results of Booth and Hall [10] and Veldsman [29]. These results have not been previously presented elsewhere to the best of our knowledge. vii In Chapter 4, the equiprime near-rings are shown to yield an ideal-hereditary radical in N∘. It is shown that a special radical theory can be built on the equiprime nearrings in much the same way prime rings are used in ring theory to define special radical classes of rings.

In view of the facts that the definition of a ring led to the definition of a near- ring, the definition of a ring module led to the definition of a near-ring module, prime rings resulted in investigations with respect to primeness in near-rings, one is naturally inclined to attempt to define the notion of a group near-ring seeing that the group ring had already been defined and investigated into by, interalia, Groenewald in [7] . However, in trying to define the group near-ring along the same lines as the group ring was defined, it was found that the resulting multiplication was, in general, not associative in the near-ring case due to the lack of one distributive property. In 1976, Meldrum [19] achieved success in defining the group near-ring. How- ever, in his definition, only distributively generated near-rings were considered and the distributive generators played a vital role in the construction. In 1989, Le Riche, Meldrum and van der Walt [17], adopted a similar approach to that which led to a successful and fruitful definition of matrix near-rings, and defined the group near-ring in a more general sense. In particular, they defined R[G], the group near-ring of a group G over a near-ring R, as a subnear-ring of M(RG), the near-ring of all mappings of the group RG into itself. More recently, Groenewald and Lee [14], further generalised the definition of R[G] to R[S : M], the generalised semigroup near-ring of a semigroup S over any faithful R-module M. Again, the natural thing to do would be to extend the results obtained for R[G] to R[S : M], and this they achieved with much success.