# near-ring

## Definitions

A near-ring is a set (http://planetmath.org/Set) $N$ together with two binary operations, denoted $+\colon N\times N\to N$ and $\cdot\colon N\times N\to N$, such that

1. 1.

$(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ for all $a,b,c\in N$ (associativity of both operations)

2. 2.

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

3. 3.

For all $a\in N$, there exists $b\in N$ such that $a+b=b+a=0$ (additive inverse)

4. 4.

$(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ for all $a,b,c\in N$ (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 commutative (http://planetmath.org/Commutative), and only require distributivity on one side.

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

## Notes

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

We say $N$ has an if there exists an element $1\in N$ such that $a\cdot 1=1\cdot a=a$ for all $a\in N$. We say $N$ is distributive if $a\cdot(b+c)=(a\cdot b)+(a\cdot c)$ holds for all $a,b,c\in N$. We say $N$ is commutative if $a\cdot b=b\cdot a$ for all $a,b\in N$.

Every commutative near-ring is distributive. Every distributive near-ring with an identity element is a unital ring (see the attached proof (http://planetmath.org/ConditionOnANearRingToBeARing)).

## Example

A natural example of a near-ring is the following. Let $(G,+)$ be a group (not necessarily abelian (http://planetmath.org/AbelianGroup2)), and let $M$ be the set of all functions from $G$ to $G$. For two functions $f$ and $g$ in $M$ define $f+g\in M$ by $(f+g)(x)=f(x)+g(x)$ for all $x\in G$. Then $(M,+,\circ)$ is a near-ring with identity, where $\circ$ denotes composition of functions.

## References

• 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