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.

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