Near-ring

In mathematics, a near-ring is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.

Definition

A set N together with two binary operations + and ⋅ is called a near-ring if:

N is a group under addition;
multiplication is associative ; and
multiplication on the right distributes over addition: for any x, y, z in N, it holds that ⋅z = +.

Similarly, it is possible to define a left near-ring by replacing the right distributive law by the corresponding left distributive law. Both right and left near-rings occur in the literature; for instance, the book of Pilz uses right near-rings, while the book of Clay uses left near-rings.
An immediate consequence of this one-sided distributive law is that it is true that 0⋅x = 0 but it is not necessarily true that x⋅0 = 0 for any x in N. Another immediate consequence is that ⋅y = − for any x, y in N, but it is not necessary that x⋅ = −. A near-ring is a rng if and only if addition is commutative and multiplication is also distributive over addition on the left. If the near-ring has a multiplicative identity, then distributivity on both sides is sufficient, and commutativity of addition follows automatically.

Mappings from a group to itself

Let G be a group, written additively but not necessarily abelian, and let M be the set of all functions from G to G. An addition operation can be defined on M: given f, g in M, then the mapping from G to G is given by for all x in G. Then is also a group, which is abelian if and only if G is abelian. Taking the composition of mappings as the product ⋅, M becomes a near-ring.
The 0 element of the near-ring M is the zero map, i.e., the mapping which takes every element of G to the identity element of G. The additive inverse −f of f in M coincides with the natural pointwise definition, that is, for all x in G.
If G has at least two elements, then M is not a ring, even if G is abelian. However, there is a subset E of M consisting of all group endomorphisms of G, that is, all maps such that for all x, y in G. If is abelian, both near-ring operations on M are closed on E, and is a ring. If is nonabelian, E is generally not closed under the near-ring operations; but the closure of E under the near-ring operations is a near-ring.
Many subsets of M form interesting and useful near-rings. For example:

The mappings for which.
The constant mappings, i.e., those that map every element of the group to one fixed element.
The set of maps generated by addition and negation from the endomorphisms of the group. If G is abelian then the set of endomorphisms is already additively closed, so that the additive closure is just the set of endomorphisms of G, and it forms not just a near-ring, but a ring.

Further examples occur if the group has further structure, for example:

The continuous mappings in a topological group.
The polynomial functions on a ring with identity under addition and polynomial composition.
The affine maps in a vector space.

Every near-ring is isomorphic to a subnear-ring of M for some G.

Applications

Many applications involve the subclass of near-rings known as near-fields; for these see the article on near-fields.
There are various applications of proper near-rings, i.e., those that are neither rings nor near-fields.
The best known is to balanced incomplete block designs using planar near-rings. These are a way to obtain difference families using the orbits of a fixed-point-free automorphism group of a group. James R. Clay and others have extended these ideas to more general geometrical constructions.