A setN together with two binary operations + and ⋅ is called a near-ring if: Similarly, it is possible to define a left near-ring by replacing the right distributive law A3 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 ringif and only ifaddition 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.
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 f + g from G to G is given by = f + g 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 2 elements, M is not a ring, even if G is abelian. However, there is a subset E of Mconsisting of all group endomorphisms of G, that is, all maps f : G → G such that f = f + f 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 f = 0.
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:
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 freeautomorphism group of a group. Clay and others have extended these ideas to more general geometrical constructions.
