near-ring

Definitions

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

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

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

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 identity element 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.