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:
If you try to non-linearize
you will find the near-rings nice.
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]).