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.

Algebraic structures
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space
Algebra-like Algebra Associative Non-associative Lie algebra Graded Bialgebra

Definition

A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:

A1: N is a group (not necessarily abelian) under addition;

A2: multiplication is associative (so N is a semigroup under multiplication); and

A3: multiplication distributes over addition on the right: 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 A3 by the corresponding left distributive law. However, near-rings are almost always written as right 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 distributive over addition on the left.

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; 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:

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^[2] 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^[3]

References

1 2 G. 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.
↑ G. Pilz " Near-rings", North-Holland, Amsterdam, second edition, (1983).
↑ J. Clay "Nearrings: Geneses and applications", Oxford, (1992).

Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. ISBN 978-1-4613-0267-4.

External links

The Near Ring Main Page at the Johannes Kepler Universität Linz

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