condition on a near ring to be a ring

Every ring is a near-ring. The converseMathworldPlanetmath is true only when additional conditions are imposed on the near-ring.

Theorem 1.

Let (R,+,) be a near ring with a multiplicative identityPlanetmathPlanetmath 1 such that the also left distributes over +; that is, c(a+b)=ca+cb. Then R is a ring.

In short, a distributive near-ring with 1 is a ring.

Before proving this, let us list and prove some general facts about a near ring:

  1. 1.

    Every near ring has a unique additive identity: if both 0 and 0 are additive identities, then 0=0+0=0.

  2. 2.

    Every element in a near ring has a unique additive inverse. The additive inverse of a is denoted by -a.


    If b and c are additive inverses of a, then b+a=0=a+c and b=b+0=b+(a+c)=(b+a)+c=0+c=c. ∎

  3. 3.

    -(-a)=a, since a is the (unique) additive inverse of -a.

  4. 4.

    There is no ambiguity in defining “subtraction- on a near ring R by a-b:=a+(-b).

  5. 5.

    a-b=0 iff a=b, which is just the combinationPlanetmathPlanetmath of the above three facts.

  6. 6.

    If a near ring has a multiplicative identity, then it is unique. The proof is identical to the one given for the first Fact.

  7. 7.

    If a near ring has a multiplicative identity 1, then (-1)a=-a.


    a+(-1)a=1a+(-1)a=(1+(-1))a=0a=0. Therefore (-1)a=-a since a has a unique additive inverse. ∎

We are now in the position to prove the theorem.


Set r=a+b and s=b+a. Then

r-s =r-(b+a)   substitution
=r+(-1)(b+a)   by Fact 7 above
=r+((-1)b+(-1)a)   by left distributivity
=r+(-b+(-a))   by Fact 7 above
=(a+b)+(-b+(-a))   substitution
=((a+b)+(-b))+(-a)   additive associativity
=(a+(b+(-b))+(-a)   additive associativity
=(a+0)+(-a)   -b is the additive inverse of b
=a+(-a)   0 is the additive identity
=0   same reason as above

Therefore, a+b=r=s=b+a by Fact 5 above. ∎

Title condition on a near ring to be a ring
Canonical name ConditionOnANearRingToBeARing
Date of creation 2013-03-22 17:19:54
Last modified on 2013-03-22 17:19:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Theorem
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Related topic UnitalRing