nearring
Definitions
A nearring 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 nearring 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 nearfield is a nearring $N$ such that $(N\setminus \{0\},\cdot )$ is a group.
Notes
Every element $a$ in a nearring 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 nearring is distributive. Every distributive nearring with an identity element is a unital ring (see the attached proof (http://planetmath.org/ConditionOnANearRingToBeARing)).
Example
A natural example of a nearring 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 nearring with identity^{}, where $\circ $ denotes composition of functions.
References
 1 Günter Pilz, NearRings, NorthHolland, 1983.
Title  nearring 
Canonical name  Nearring 
Date of creation  20130322 13:25:12 
Last modified on  20130322 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 nearring 
Defines  commutative near ring 
Defines  commutative nearring 
Defines  distributative nearring 
Defines  distributative near ring 
Defines  distributative nearring 
Defines  near field 
Defines  nearfield 