Every ring has both left and right modules. In the theory of nearrings, only right modules are usually considered for left nearrings. The purpose of this report is to promote the study of an alternative type of nearring module. For left nearrings, these unusual modules are left modules. There are three reasons for studying left modules for left nearrings. These unusual nearring modules add an element of symmetry to the theory of nearrings. At the same time, comparing left and right modules of a left nearring illustrates how the theory of nearrings is distinct from ring theory. Finally, with two types of nearring modules, it is possible to carry over to nearring theory more concepts from ring theory; for example, duals of modules and bimodules. This report is an attempt to show that these reasons are valid. The first chapter is devoted to producing a well-reasoned definition for the unusual type of nearring module. It begins with a careful presentation of background material on nearrings, rings, and ring modules. This material is used to motivate the definitions for nearring modules, which are introduced in the third section. The second chapter is concerned with showing that the unusual type of nearring module can fit into the theory of nearrings. In the first section, several papers relevant to the study of these modules are summarized. The work of A. Frohlich on free additions is of primary importance. General construction methods for both types of nearring modules are then described. Finally, some general properties of left modules of left nearrings are examined. Examples of left modules for left nearrings are presented in the third chapter. First, the general constructions of the second chapter are applied in some particular cases. This leads naturally to structures that are analogous to bimodules and structures analogous to dual modules for ring modules. Here, free additions have a special role. Several dual nearring modules are examined in detail. The information needed to construct many more examples of nearring modules of the unusual type is also presented. Only small cyclic groups are used for these examples.

by Lam Che Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 85-87). / Chapter 1 --- Preliminaries --- p.2 / Chapter 1.1 --- Introduction --- p.2 / Chapter 1.2 --- Ideals of Γ-nearrings --- p.6 / Chapter 1.3 --- Pierce-decomposition theorem --- p.14 / Chapter 1.4 --- Left SΓ and Right RΓ-bimodules --- p.19 / Chapter 2 --- D.G. Γ-nearrings and its modules --- p.25 / Chapter 2.1 --- Distributively generated Γ-nearrings --- p.25 / Chapter 3 --- Near-rings and Automata --- p.40 / Chapter 3.1 --- Monoids of semiautomaton and automaton --- p.40 / Chapter 4 --- Derivation in Γ-nearrings --- p.66 / Chapter 4.1 --- Derivation in Γ-nearrings --- p.66 / Chapter 4.2 --- Abelian conditions --- p.70 / Chapter 4.3 --- Unitary Γ-nearrings --- p.76 / Chapter 4.4 --- Decomposition of right Rr-modules --- p.81

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.

The family of planar nearrings enjoys quite a few geometric and combinatoric properties. Circular planar nearrings are members of this family which have the character of circles of the complex plane. On the other hand, they also have some properties which one may not find among the circles of the complex plane. In this dissertation, we first review the definition and characterization of a planar nearring, and some various ways of constructing planar nearrings, as well as various ways of constructing BIBD's from a planar nearring. Circularity of a planar nearring is then introduced, and examples of circularity planar nearrings are given. Then, some nonisomorphic BIBD's arising from the same additive group of a planar nearring are examined. To provide examples of nonabelian planar nearrings, the structures of Frobenius groups with kernel of order 64 are completely determined and described. On the other hand, examples of Ferrero pairs (N, Φ)'s with nonabelian Φ, which produce circular planar nearrings, are provided. Finally, we study the structures of circular planar nearrings generated from the finite prime fields from geometric and combinatoric points of view. This study is then carried back to the complex plane. In turn, it gives a good reason for calling a block from a circular planar nearring a "circle."

The primary objective of this work is to discuss some of the elementary properties of near-rings as they are related to rings. This study is divided into three subdivisions: (1) Basic Properties and Concepts of Near-Rings; (2) The Ideal Structure of Near-Rings; and (3) Homomorphism and Isomorphism of Near-Rings.

The prototype of a near-ring is the set of all self-maps of an additively written (but not necessarily abelian) group under pointwise addition and composition of maps. Moreover, any near-ring with unity can be embedded in a near-ring (with unity) of self-maps of some group. For this reason, a lot of research has been done on near-rings of maps. In 1979, Hofer [16] gave the study of near-rings of maps a topological avour by considering the near- ring of all continuous self-maps of a topological group. In this dissertation we consider some standard constructions of near-rings of maps on a group G and investigate these when G is a topological group and our near-ring consists of continuous maps.

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 1991. / After an introduction defining basic structutral aspects of near-rings, this report examines how the ring-theoretic notions of generation and cogeneration can be extended from modules over a ring to modules over a near-ring. Chapter four examines matrix near-rings and connections between the J2 and JS radicals of the near-ring and the corresponding matrix near-ring. By extending the concepts of generation and cogeneration from the ring modules to near-ring modules we are investigating how important distribution and an abelian additive structure are to these two concepts. The concept of generation faces the obstacle that the image of a near-ring module homomorphism is not necessarily a subrnodule of the image space but only a subgroup, while the sum of two subgroups need not even be a subgroup. In chapter two, generation trace and socle are defined for near-ring modules and these ideas are linked with those of the essential and module-essential subgroups. Cogeneration, dealing with kernels which are always submodules proved easier to generalise. This is discussed in chapter three together with the concept of the reject, and these ideas are Iinked to the J1/2 and J2 radicals. The duality of the ring theory case is lost. The results are less simple than in the ring theory case due to the different types of near-ring module substructures which give rise to several Jacobson-type radicals. A near-ring of matrices can be obtained from an arbitrary near-ring by regarding each rxr matrix as a mapping from Nr to Nr where N is the near-ring from which entries are taken. The argument showing that the near-ring is 2-semisimple if and only if the associated near-ring of matrices is 2-semisimple is presented and investigated in the case of s-semisimplicity. Questions arising from this report are presented in the final chapter.

Student Number : 9206749J - PhD thesis - School of Mathematics - Faculty of Science / For right near-rings the left representation has always been considered the natural one. A study of right representation for right distributively generated (d.g.) near-rings was initiated by Rahbari and this work is extended here to introduce radical-like objects in the near-ring R using right R-groups. The right radicals rJ0(R), rJ1/2(R) and rJ2(R) are defined as counterparts of the left radicals J0(R), J1/2(R) and J2(R) respectively, and their properties are discussed. Of particular interest are the relationships between the left and right radicals. It is shown for example that for all finite d.g. near-rings R with identity, J2(R) = rJ0(R) = rJ1/2(R) = rJ2(R). A right anti-radical, rSoi(R), is defined for d.g. near-rings with identity, using a construction that is analogous to that of the (left) socle-ideal, Soi(R). In particular, it is shown that for finite d.g. near-rings with identity, an ideal A is contained in rSoi(R) if and only if A \ J2(R) = (0). The relationship between the left and right socle-ideals is investigated, and it is established that rSoi(R) #18; Soi(R) for d.g. near-rings with identity and satisfying the descending chain condition for left R-subgroups.

La memoria trata algunos aspectos de la teoría de casi-anillos de polinomios r(x) con coeficientes en un anillo r conmutativo y con unidad. En el capítulo I damos una descripción explicita de los elementos distributivos de r(x) y de la parte cero-simétrica r sub 0 (x). En los párrafos damos algunas caracterizaciones y propiedades del anillo formado por estos elementos distributivos. Obtenemos resultados similares en el casi-anillo de series de potencias formales. En el capítulo II está dedicado al estudio de subcasi-anillos que gozan de las dos propiedades distributivas en r (x) y de ideales de casi-anillos que dan cociente anillo particularizando esto para el caso del casi-anillo r(x). En el capítulo III encontramos todos los ideales maximales de z (x) (z el anillo de los enteros). Estudiamos también los ideales de composición del anillo de composición (r(x) + o) dando una descripción de todos los maximales. Acaba la memoria con un algoritmo para la descomposición de polinomios con coeficientes en cuerpo f es decir encontramos una descomposición de un polinomio en componentes indescomponibles / In this dissertation we study several aspects of near-rings. In the first chapter we give an explicit description of the distributive elements of the near-ring of polynomials R[x], over a commutative ring R a with identity. We also find the distributive elements in the near-ring of formal power series over a commutative rings with identity. In the second chapter, we search rings which are contained in R[x], we prove that if R is an integral domain, the set of distributive elements contains the subrings of the near-rings of polynomials. We also investigate ideals I of the near-ring such that the quotient is ring. In the next chapter we find all maximal ideals in Z[x] and maximal full ideals in the composition rings. The last section we provide the first polynomial time algorithm for decomposing polynomials into indecomposable ones.