at least 2 elements, M(G) is not a ring, even if G is abelian. (Consider a constant mapping g from G to a fixed element g ≠ 0 of G; then g⋅0 = g ≠ 0.) 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. If (G, +) is nonabelian, E(G) is generally not closed under the near-ring operations; but the closure of E(G) under the near-ring operations is a near-ring. Many subsets of M(G) form interesting and useful near-rings. For example: The mappings for which f(0) = 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 (the "additive closure" of the set of endomorphisms). 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(G) 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. Clay and others have extended these ideas to more general geometrical constructions. See also Near-field (mathematics) Semiring Near-semiring References External links The Near Ring Main Page at the Johannes Kepler Universität Linz Algebraic structures Ring theory