A near-ring is a set ( N together with two binary operationsMathworldPlanetmath, denoted +:N×NN and :N×NN, such that

  1. 1.

    (a+b)+c=a+(b+c) and (ab)c=a(bc) for all a,b,cN (associativity of both operationsMathworldPlanetmath)

  2. 2.

    There exists an element 0N such that a+0=0+a=a for all aN (additive identity)

  3. 3.

    For all aN, there exists bN such that a+b=b+a=0 (additive inverse)

  4. 4.

    (a+b)c=(ac)+(bc) for all a,b,cN (right distributive law)

Note that the axioms of a near-ring differ from those of a ring in that they do not require addition to be commutativePlanetmathPlanetmathPlanetmath (, and only require distributivity on one side.

A near-field is a near-ring N such that (N{0},) is a group.


Every element a in a near-ring has a unique additive inverse, denoted -a.

We say N has an identity elementMathworldPlanetmath if there exists an element 1N such that a1=1a=a for all aN. We say N is distributive if a(b+c)=(ab)+(ac) holds for all a,b,cN. We say N is commutative if ab=ba for all a,bN.

Every commutative near-ring is distributive. Every distributive near-ring with an identity element is a unital ring (see the attached proof (


A natural example of a near-ring is the following. Let (G,+) be a group (not necessarily abelian (, and let M be the set of all functions from G to G. For two functions f and g in M define f+gM by (f+g)(x)=f(x)+g(x) for all xG. Then (M,+,) is a near-ring with identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, where denotes composition of functions.


  • 1 Günter Pilz, Near-Rings, North-Holland, 1983.
Title near-ring
Canonical name Nearring
Date of creation 2013-03-22 13:25:12
Last modified on 2013-03-22 13:25:12
Owner yark (2760)
Last modified by yark (2760)
Numerical id 22
Author yark (2760)
Entry type Definition
Classification msc 16Y30
Synonym near ring
Synonym nearring
Related topic Ring
Defines commutative near-ring
Defines commutative near ring
Defines commutative nearring
Defines distributative near-ring
Defines distributative near ring
Defines distributative nearring
Defines near field
Defines nearfield