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Near-rings: What they are and what they are good for

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by Gunter Pilz in Linz, Austria

Near-rings are generalised rings. They might generally be described as
rings (N,+,*) where the addition is not necessarily
abelian and only one distributive law holds. Or they can be
defined axiomatically.
Near-rings arise naturally in studying functions on a group
(G,+).
M(G) = (f:G->G,+,o)
is then a near-ring which is not a ring.
E(G) the additive group
generated by all the endomorphisms of G is an important
subnear-ring of M(G) which generalises the concept of of an endomorphism
ring on an abelian group to the non-abelian case.
Another class of examples is provided by polynomials w.r.t. addition and
substitution.

Near-rings in which the nonzero elements form a multiplicative
group are called near-fields. They were
introduced at the beginning of this century and soon proved useful in coordinatising certain important classes of geometric planes.
Later on, they also turned out to be an essential tool in studying
doubly transitive groups and incidence structures.

The study of "proper" near-rings, on the other hand,
culminated in theorems
which considerably generalise Wedderburn-Artin's Theorem and
Jabobson's Density Theorem in ring theory.
For near-rings, density is equivalent to an interesting
interpolation property, since one is dealing with non-linear
functions (in contrast to the "linear situation" in
ring theory). Also, these structure theorems give answers to the questions,
when every function on an omega group is (or can be) interpolated
by a polynomial function.
Polynomial near-rings turn out to be very useful in determining generalized
ideals in omega groups.
Finally, a class of near-rings, the "planar" ones, give rise to
Balanced Incomplete Block Designs, structures which are needed
for experimental designs.
From planar near-rings, Block Designs with extra-ordinary
high efficiencies can be constructed.

If this is a bit too abstract, then a collection of
algebraic structures that includes definitions and
all has been compiled by John Pederson
here.
This is quite an interesting and useful resource,
and will hopefully continue to grow.