nearring

A near-ring or nearring is an algebraic structure with 2 binary operations arising natually from a group. Let

(G,+)

be a group, which may be nonabelian, and let

M(G)=\{f\mid f:G\to G\}

, the set of all mappings from

G

G

. An addition can be defined on

M(G)

: if

f,g\in M(G)

, then the mapping

f+g:G\to G

is given by

(f+g)(x)=f(x)+g(x)

for all

x\in G

. Then

(M(G),+)

is a group, which is abelian if and only if

G

is. Taking the composition of mappings as the product,

(M(G),+,\circ)

becomes a prototypal nearring. One notices that

(M(G),+)

is a group which may not be abelian,

(M(G),\circ)

is a semigroup,

for any

f,g,h\in M(G)

(f+g)\circ h=f\circ  h+g\circ h

but it is in general not true that

h(f+g)=h\circ f+h\circ g

(thus only one side distributive law holds).

An immediate consequence of this one-sided distributive law is that it is true that

0\circ f=0

but it may also be true that

f\circ 0\not=0

for an

f\in M(G)

. Here

0

denotes the 0-map, i.e. the mapping which takes every element of

G

to the zero element of

G

. Now, one can define abstract nearrings as follows. A set

N

together with two binary operations

+

and

\cdot

is called a (right) nearring if

(N,+)

is a (not necessarily abelian) group,

(N,\cdot)

is a semigroup,

for any

x,y,z\in N

, it holds that

(x+y)\cdot z=x\cdot z+y\cdot z

It is true that all rings are nearrings, but not the converse as the example

M(G)

given at the beginning of this article shows.

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