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Near-Rings and Near-Fields: Proceedings of a Conference Held at the University of Tubingen, F.R.G. 4-10 August, 1985 (North-Holland Mathematics Studies) By Gerhard Betsch

INTRODUCTION

By Gerhard Betsch

The papers in this volume provide a fairly complete picture of current trends and problems in the

theory of near-rings and near-fields. In the sequel we shall describe a few condensation points. But

first we shall mention some

Basic defilunons

A

that

(i)

(ii) multiplication is associative;

(iii) (a+b)c=ac+bc for all a,b,ce N.

A leftnear-ring is defined in an analogous manner: (iii) has to be replaced by the other distributive

law

(iii') c(a+b)=ca+cb for all a,b,ce N.

In the sequel we shall - with few exceptions - simply speak of "near-rings", because frequently it is

inessential for the purpose of this introduction, which type of near-rings (left or left) is

investigated in the paper under discussion.

. .

is a triple (N,+,.), where N is a set and + and . are binary operations on N such

(N,+) is a group (not necessarily commutative);

Examples of near-rings are abundant: They arise in a natural way whenever one deals with systems

of "non-linear" mappings. E.g., let (r',+)b e a not necessarily commutative group. Then all

mappings of r into itself (written as left operators of r) form a (left) near-ring M(T) with repect to

pointwise addition and multiplication by composition. In most cases, M(T) has many

subnear-rings. Conversely, it can be proved, that any (left) near-ring N may be embedded into a

suitable near-ring M(T).

A near-ring N is called m o - s v 'm if nO=On=O holds for all nE N (0 is the neutral element of

the additive group (N,+,)).

A near-field is a near-ring N with the property that (N\(O),.) is a group.

All near-fields with more than 2 elements are zero-symmetric. The additive group of a near-field is

commutative [ I I , Prop. 8.1 and Theorem 8.111.

(Numbers in square brackets refer to the short bibliography at the end of this introduction, while

names in capital letters refer to the paper of the quoted author(s) in this volume.)

The &,& of a near-ring N may be defined as the kernels of (near-ring) homomorphisms of N. If N

is a left near-ring, then an N-group the near-ring analogue of a left module over a ring. (N,+)is

an N-group by left multiplication. The of N are precisely the kernels of N-group



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