In this paper, using the idea of upper and lower faithful d.g. near-rings, introduced in (6), we show that the category of all faithful d.g. near-rings is a reflective as well as coreflective subcategory of the category of all d.g. near-rings. We also prove that both and are complete and cocomplete categories.

A permutation group G operating on a set M is called a ZT-group (Zassenhaus transitive group) if G has the properties (i) and (ii):

(i) G operates 2-transitively on M;

(ii) Ga,b≠{id} and Ga,b,c = {id} for distinct elements a,b,c ∈ M.

Here Ga,b = {α ∈ G | α(a) = a and α(b) = b} denotes the stabilizer of {a, b}, and Ga,b,c the stabilizer of {a, b, c}, respectively.

In somes categories, there are structures that look very much like groups, and they usually are. These structures are called group-objects and were first studied by Eckmann and Hilton (1). If our category has an object T such that hom(X, T)= {tx}, a singleton, for each object X ∈ Ob , T is called a terminal object. Our category must have products; i.e. for A1,…, An ∈;. Ob , there is an object A1 × … × An ∈ Ob and morphisms pi: A1 × … × An → Ai so that if fi: X → Ai, i = 1, 2, …, n, are morphisms of , then there is a unique morphism [f1, …, fn]: X → A1 × … × An such that for i = 1, 2, …, n.

A strongly regular ring R is one in which for all x ∈ R, there is an a ∈ R with x = x2a. Equivalently, for all x there is an a with x = ax2. Such a ring is regular, duo, biregular, and a left and right V-ring. Moreover since R is reduced, all nilpotent elements are central (vacuously) and so all idempotent elements are central.

In this paper radicals in the sense of Kuroš and Amitsur (KA-radicals) for Ω-groups will be studied. For the sake of simplicity these radicals will be considered on varieties, the results remaining valid for more general classes.

Some techniques developed by Eilenberg (1) in the theory of automata are used to investigate certain types of near-ring.

Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp < S; > where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.

Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.

Let R be an arbitrary near-ring and define the multiplicative centre Z(R) by

In previous papers (2,3,5) we have established additive or multiplicative commutativity for various near-rings R in which selected elements were restricted to lie in Z(R); the near-rings involved were usually distributively-generated (d-g) and were frequently assumed to have a multiplicative identity element as well.

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.

In this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance, M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). If N is a near-ring then N0: = {n ∈ N/n0 = 0}. A group (Γ, +) is an N-group (we write NΓ) if a “product” ny is defined with (n + n‛)γ = nγ + n‛γ and (nn‛)γ = n(n‛γ). Ideals of near-rings and N-groups are kernels of (N-) homomorphisms. If Γ is a vector-space, Maff (Γ) is the near-ring of all affine transformations on Γ. N is 2-primitive on NΓ if NΓ is non-trivial, faithful and without proper N-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.

Let N be an arbitrary near-ring. Each element a ∈ N determines in a natural way a new multiplication on the elements of N which results in a near-ring Na whose additive group coincides with that of N but whose multiplicative semigroup generally differs. Specifically, we define the product x * y of two elements in Na by x * y = x a y where a product in the original near-ring is denoted by juxtaposition. One easily checks that Na is a near-ring with addition identical to that of N. The original near-ring N will be referred to as the base near-ring, Na will be referred to as a laminated near-ring of N and a will be referred to as the laminating element or sometimes more simply as the laminator.

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

Throughout this note, N will denote a (Left) near-ring with two-sided zero. Definitions of basic concepts can be found in (9).

We prove first that a right ideal I in a d.g. near-ring has a right identity if and only if x ∈ xI for each x ∈I. This enables us to study the structure of regular d.g. near-rings with chain conditions on right annihilators. Specifically we will prove that a regular d.g. near-ring with both the maximum and the minimum conditions on right annihilators is a finite direct sum of near-rings which are either rings of matrices over division rings or non-rings of the form MG(Γ) for a suitable type 2 N-module Γ. Finally we consider the case of maximum condition on N-subgroups.

Nearfields and near-rings are related to many other structures and needed for several representation theorems. Therefore it is important to gain knowledge about the structure of near-rings and nearfields and to find construction methods. The first examples of proper nearfields were constructed by L.E. Dickson 1905, they were finite. 30 years later H. Zassenhaus completely determined the finite nearfields his attention having been attracted to them by the study of certain permutation groups. By the axiomatization of Dickson's methods, H. Karzel succeeded in giving new examples of infinite nearfields. In the extensive generalization by J. Timm the so-called “Dicksonian processes” are the most important tool for constructions of nearfields and near-rings. The report (44) by H. Wähling is a summary of the results on nearfields obtained so far.