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.
Definition.
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. 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.
Mappings from a group to itself.
Let "G" be a group, written additively but not necessarily abelian, and let "M"("G") be the set {"f" | "f" : "G" → "G"} 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". The additive inverse −"f" of "f" in "M"("G") coincides with the natural pointwise definition, that is, (−"f")("x") = −("f"("x")) for all "x" in "G".
If "G" has 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:
Further examples occur if the group has further structure, for example:
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.