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

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.