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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