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# near-ring

# 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. 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.

# 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.

# References

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

## Mathematics Subject Classification

16Y30*no label found*

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