# 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 |
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## 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
*on the right*distributes 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 A3 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 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:[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

- 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, the Theory and its Applications", North-Holland, Amsterdam, 2nd 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.