Nearring

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In mathematics, a near-ring or nearring is an algebraic structure with two binary operations arising naturally from a group. Let <math>(G,+)</math> be a group, which may be nonabelian, and let <math>M(G)=\{f\mid f:G\to G\}</math>, the set of all mappings from <math>G</math> to <math>G</math>. An addition can be defined on <math>M(G)</math>: if <math>f,g\in M(G)</math>, then the mapping <math>f+g:G\to G</math> is given by <math>(f+g)(x)=f(x)+g(x)</math> for all <math>x\in G</math>. Then <math>(M(G),+)</math> is a group, which is abelian if and only if <math>G</math> is. Taking the composition of mappings as the product, <math>(M(G),+,\circ)</math> becomes a prototypal nearring. One notices that

<math>(M(G),+)</math> is a group which may not be abelian,

<math>(M(G),\circ)</math> is a semigroup,

for any <math>f,g,h\in M(G)</math>, <math>(f+g)\circ h=f\circ h+g\circ h</math> but it is in general not true that <math>h\circ(f+g)=h\circ f+h\circ g</math> (thus only one side distributive law holds).

An immediate consequence of this one-sided distributive law is that it is true that <math>0\circ f=0</math> but it may also be true that <math>f\circ 0\not=0</math> for an <math>f\in M(G)</math>. Here <math>0</math> denotes the 0-map, i.e. the mapping which takes every element of <math>G</math> to the zero element of <math>G</math>.

Now, one can define abstract nearrings as follows.

A set <math>N</math> together with two binary operations <math>+</math> and <math>\cdot</math> is called a (right) nearring if

<math>(N,+)</math> is a (not necessarily abelian) group,

<math>(N,\cdot)</math> is a semigroup,

for any <math>x,y,z\in N</math>, it holds that <math>(x+y)\cdot z=x\cdot z+y\cdot z</math>.

It is true that all rings are nearrings, but not the converse as the example <math>M(G)</math> given at the beginning of this article shows.