Monoids, Groups, Rings and Fields
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 5 June 2012
Monoids, Groups, Rings and Fields

A monoid without identity is a set $G$ with a function
$$\begin{array}{rcl}?:G\times G& \u27f6& G\\ (i,j)& \u27fc& i?j\end{array}$$
such that
 (a) ($?$ is associative) if
$i,j,k\in G$
then $(i?j)?k=i?(j?k)$.
 A monoid is a set $G$ with a function
$$\begin{array}{rcl}?:G\times G& \u27f6& G\\ (i,j)& \u27fc& i?j\end{array}$$
such that
 (a) ($?$ is associative) if
$i,j,k\in G$
then $(i?j)?k=i?(j?k)$, and
 (b) ($G$ has an identity) there exists an element
$!\in G$ such that if
$y\in G$ then $!?y=y?!=y$.
 A commutative monoid is a set $G$ with a function
$$\begin{array}{rcl}+:G\times G& \u27f6& G\\ (i,j)& \u27fc& i+j\end{array}$$
such that
 (a) $G$ is a monoid, and
 (b) if $i,j\in G$
then $i+j=j+i$.
 A group is a set $G$ with a function
$$\begin{array}{rcl}?:G\times G& \u27f6& G\\ (i,j& \u27fc& i?j\end{array}$$
such that
 (a) ($?$ is associative) if
$i,j,k\in G$
then $(i?j)?k=i?(j?k$,
 (b) ($G$ has an identity) there exists an element
$!\in G$ such that if
$y\in G$ then
$!?y=y?!=y$, and
 (c) ($G$ has inverses) if $y\in G$
there is an element ${y}^{\#}\in G$
such that $y?{y}^{\#}={y}^{\#}?y=!$ where $!$ is the identity in
$G$.
 An abelian group is a set $G$ with a function
$$\begin{array}{rcl}+:G\times G& \u27f6& G\\ (i,j)& \u27fc& i+j\end{array}$$
such that
 (a) $G$ is a group, and
 (b) if $i,j,k\in G$
then $i+j=j+i$.
 A ring without identity is a set $R$ with two functions
$$\begin{array}{rcl}+:R\times R& \u27f6& R\\ (i,j)& \u27fc& i+j\end{array}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\begin{array}{rcl}\times :R\times R& \u27f6& R\\ (i,j)& \u27f6& i\times j=ij\end{array}$$
such that
 (a) $R$ with $+$ is an abelian group,
 (b) $R$ with $\times $ is a monoid without identity, and
 (c) $R$ has distributive laws; if
$i,j,k\in R$
then $i(j+k)=ij+ik$
and $(i+j)k=ik+jk$.

A ring is a ring without identity $R$
such that there is an element $1\in R$ such that if
$y\in R$ then $1y=y1=y$.

A commutative ring is a ring such that if
$x,y\in R$ then
$xy=yx$.
 A field is a commutative ring $\mathbb{F}$ such that if
$y\in \mathbb{F}$ and $y\ne 0$
(the identity with respect to $+$)
then there is an element ${y}^{1}\in \mathbb{F}$ with
$y{y}^{1}={y}^{1}y=1$.

A division ring is a ring $\mathbb{D}$ such that if
$y\in \mathbb{D}$ and
$y\ne 0$ (the identity with respect to
$+$) then there is an element
${y}^{1}\in \mathbb{D}$
with $y{y}^{1}={y}^{1}y=1$.
Examples.

(a) The positive integers ${\mathbb{Z}}_{>0}$
with the addition operation is a monoid without identity.
 (b)
The nonnegative integers ${\mathbb{Z}}_{\ge 0}$
with the addition operation is a monoid.
 (c) The integers $\mathbb{Z}$ with the addition operation is an abelian group.
 (d) The integers $\mathbb{Z}$ with the addition and multiplication operations is a
commutative ring.
 (e) The rationals $\mathbb{Q}$ with the operations addition and multiplication
is a field.
 (f) The quaternions $\mathbb{H}$ with the oeprations of addition and
multiplication form a division ring that is not a field.
 (g) The $2\times 2$ matrices with entries in $\mathbb{Z}$, ${M}_{2}\left(\mathbb{Z}\right)$,
with the operations of addition and multiplication of matrices is a ring which is not commutative.
 (h) The set of invertible
$2\times 2$ matrices with entries in $\mathbb{Z}$, ${\mathrm{GL}}_{2}\left(\mathbb{Z}\right)$, with
the operation of multiplication of matrices is a group which is not an abelian group.
Notes and References
Welcome to our (algebraic) zoo.
This page tells you the types of animals in our zoo.
The unusual notations $?$, $!$,
and ${y}^{\#}$ used in the above definitions
of monoid and group are there to make the point that the usual notations (of
$i+j$ for addition of $i$
and $j$,
$ij$ for multiplication of $i$ and
$j$, of $0$ for the additive identity,
of $1$ for the mutliplicative identity, of
$y$ for the additive inverse, and of
${y}^{1}$ for the multiplicative
inverse of $y$) are rather arbitrary. But habits are important and
useful and often efficient and helpful for clear communication,
and so it is better to use the standard notations unless there is a very good reason not to.
These definitions are found in [Bou, Alg. Ch. I].
References
[Bou]
N. Bourbaki,
Algebra, Masson?????
MR?????.
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