multiplication on the rightdistributes over addition: for any x, y, z in N, it holds that (x + y)⋅z = (x⋅z) + (y⋅z).[1]
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[2] uses right near-rings, while the book of Clay[3] 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 rngif 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 two elements, then M(G) is not a ring, even if G is abelian. (Consider a constant mappingg 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:[1]
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:
^ abG. Pilz, (1982), "Near-Rings: What They Are and What They Are Good For" in Contemp. Math., 9, pp. 97–119. Amer. Math. Soc., Providence, R.I., 1981.
^ abJ. Clay, "Nearrings: Geneses and applications", Oxford, (1992).
Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. ISBN978-1-4613-0267-4.