In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

but satisfying fewer axioms. Near-rings arise naturally from functions on

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

Definition

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

(not necessarily abelian) under addition; * multiplication is associative (so ''N'' is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

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

(not necessarily with unity)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...

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

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s 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 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...

, 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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...

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

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...

s 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

s 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 In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

. *The polynomial functions on a ring with identity under addition and polynomial composition. *The affine maps in a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

. 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.

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

Definition

Mappings from a group to itself

Applications

See also

References

External links