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 rng 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 from ''G'' to ''G'' is given by 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, 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 mapping ''g'' from ''G'' to a fixed element of ''G''; then .) However, there is a subset ''E''(''G'') of ''M''(''G'') consisting of all group endomorphisms of ''G'', that is, all maps such that for all ''x'', ''y'' in ''G''. If is abelian, both near-ring operations on ''M''(''G'') are closed on ''E''(''G''), and is a ring. If 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 .
* 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. James R. 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