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:
* ''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'').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.

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 PilzG. Pilz,
Near-rings, the Theory and its Applications
, North-Holland, Amsterdam, 2nd edition, (1983). uses right near-rings, while the book of ClayJ. Clay, "Nearrings: Geneses and applications", Oxford, (1992). 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 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:
*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 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.

See also

* Near-field (mathematics)
* Semiring
* Near-semiring

References

* {{cite book, author1=Celestina Cotti Ferrero, author2=Giovanni Ferrero, title=Nearrings: Some Developments Linked to Semigroups and Groups, year=2002, publisher=Kluwer Academic Publishers, isbn=978-1-4613-0267-4

External links

* Th
Near Ring Main Page
at the Johannes Kepler Universität Linz

Algebraic structures
Ring theory