Near-ring

In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups. A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if: N is a group (not necessarily abelian) under addition; multiplication is associative (so N is a semigroup under multiplication); and multiplication on the right distributes over addition: for any x, y, z in N, it holds that (x + y)⋅z = (x⋅z) + (y⋅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 (−x)⋅y = −(x⋅y) for any x, y in N, but it is not necessary that x⋅(−y) = −(x⋅y). A near-ring is a ring (not necessarily with unity) 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. Let G be a group, written additively but not necessarily abelian, and let M(G) be the set of all functions from G to G. An addition operation can be defined on M(G): given f, g in M(G), then the mapping f + g from G to G is given by (f + g)(x) = f(x) + g(x) for all x in G. Then (M(G), +) is also a group, which is abelian if and only if G is abelian. Taking the composition of mappings as the product ⋅, M(G) becomes a near-ring. The 0 element of the near-ring M(G) is the zero map, i.e., the mapping which takes every element of G to the identity element of G.