Definitions

A near-ring is a 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. $(a+b)+c = a+(b+c)$ 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, 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).

Example

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 $f+g\in M$ 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.

Bibliography

1

Günter Pilz, Near-Rings, North-Holland, 1983.