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 $$\begin{array}{rcl}?:G\times G& ⟶& G\\ (i,j)& ⟼& i?j\end{array}$$ such that
  1. (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 $$\begin{array}{rcl}?:G\times G& ⟶& G\\ (i,j)& ⟼& i?j\end{array}$$ such that
  1. (a) ($?$ is associative) if $i,j,k\in G$ then $(i?j)?k=i?(j?k)$, and
  2. (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 $$\begin{array}{rcl}+:G\times G& ⟶& G\\ (i,j)& ⟼& i+j\end{array}$$ such that
  1. (a) $G$ is a monoid, and
  2. (b) if $i,j\in G$ then $i+j=j+i$.
• A group is a set $G$ with a function $$\begin{array}{rcl}?:G\times G& ⟶& G\\ (i,j& ⟼& i?j\end{array}$$ such that
  1. (a) ($?$ is associative) if $i,j,k\in G$ then $(i?j)?k=i?(j?k$,
  2. (b) ($G$ has an identity) there exists an element $!\in G$ such that if $y\in G$ then $!?y=y?!=y$, and
  3. (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 $$\begin{array}{rcl}+:G\times G& ⟶& G\\ (i,j)& ⟼& i+j\end{array}$$ such that
  1. (a) $G$ is a group, and
  2. (b) if $i,j,k\in G$ then $i+j=j+i$.
• A ring without identity is a set $R$ with two functions $$\begin{array}{rcl}+:R\times R& ⟶& R\\ (i,j)& ⟼& i+j\end{array}\text{and}\begin{array}{rcl}\times :R\times R& ⟶& R\\ (i,j)& ⟶& i\times j=ij\end{array}$$ such that
  1. (a) $R$ with $+$ is an abelian group,
  2. (b) $R$ with $\times$ is a monoid without identity, and
  3. (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 $𝔽$ such that if $y\in 𝔽$ and $y\ne 0$ (the identity with respect to $+$) then there is an element $y^{-1}\in 𝔽$ with $y{y}^{-1}=y^{-1}y=1$.
• A division ring is a ring $𝔻$ such that if $y\in 𝔻$ and $y\ne 0$ (the identity with respect to $+$) then there is an element $y^{-1}\in 𝔻$ with $y{y}^{-1}=y^{-1}y=1$.

Examples.

1. (a) The positive integers ${\mathbb{Z}}_{>0}$ with the addition operation is a monoid without identity.
2. (b) The nonnegative integers ${\mathbb{Z}}_{\ge 0}$ with the addition operation is a monoid.
3. (c) The integers $\mathbb{Z}$ with the addition operation is an abelian group.
4. (d) The integers $\mathbb{Z}$ with the addition and multiplication operations is a commutative ring.
5. (e) The rationals $\mathbb{Q}$ with the operations addition and multiplication is a field.
6. (f) The quaternions $ℍ$ with the oeprations of addition and multiplication form a division ring that is not a field.
7. (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.
8. (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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