A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if: Similarly, it is possible to define a left near-ring by replacing the right distributive law by the corresponding left distributive law.
If the near-ring has a multiplicative identity, then distributivity on both sides is sufficient, and commutativity of addition follows automatically.
However, there is a subset E(G) of M(G) consisting of all group endomorphisms of G, that is, all maps f : G → G such that f(x + y) = f(x) + f(y) for all x, y in G. If (G, +) is abelian, both near-ring operations on M(G) are closed on E(G), and (E(G), +, ⋅) is a ring.
The best known is to balanced incomplete block designs[2] using planar near-rings.
James R. Clay and others have extended these ideas to more general geometrical constructions.