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### Near-ring theory

A near-ring is an algebra , where (F,+) is a (not necessarily abelian) group and the distributive law holds. Near-rings are a powerful tool in the following branches of mathematics:

• Geometry: A special class of near-rings, namely near-fields, have been used for coordinatizing a class of non-desarguesian projective planes. [Cla92] describes methods for constructing BIB-designs from near-rings. The incidence matrices of these matrices are the basis for good codes.
• Functions: The set of all selfmaps of a group G forms a near-ring if one defines addition pointwise and multiplication as functional composition. Near-ring theory is a powerful framework for studying algebraic interpolation problems, such as the polynomial or affine completeness of an -group. Famous theorems in ring theory, as e.g. Jacobson's density theorem, have their near-ring counterparts and generalizations, which make it obvious that, whereas ring-theory allows the study of linear functions, near-ring-theory is the right context to study non-linear functions, or as some people put it:
If you try to non-linearize
you will find the near-rings nice.
A recent paper by S.D.Scott ([Sco95b]) shows that near-rings can be used as a frame-work to develop a general theory of -groups.

While reading the literature and working on central questions of near-ring theory, as for example the question whether the centralizer near-ring has a distributive lattice of left ideals, I realized that for advances in the theory it is indispensible to compute interesting examples of function near-rings. A similar problem is the question whether there exist 2-tame near-rings (cf. [Sco80]) that are not 3-tame. Although at first glance these problems might seem a peculiar hobby of some mathematicians (actually, though, there are about 30 groups all over the world who work actively on near-rings), a closer inspection shows that answers to these questions will have an important impact on near-ring research: The distributivity of the lattice of left ideals is not only an interesting question by itself, but implies an interesting density-property for this near-ring (cf. [Aic95a]).

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Juergen Ecker
Tue Jan 7 14:51:38 MET 1997