Groups, Rings and Fields

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 12 October 2009

Monoids, groups, rings and fields

A monoid without identity is a set $G$ with an operation $$\begin{array}{rcl}?:G\times G& ⟶& G\\ \left(i,j\right)& ⟼& i?j\end{array}$$ such that

1. $?$ is associative; if $i,j,k\in G$ then $\left(i?j\right)?k=i?\left(j?k\right)$.

A monoid is a set $G$ with an operation $$\begin{array}{rcl}?:G\times G& ⟶& G\\ \left(i,j\right)& ⟼& i?j\end{array}$$ such that

1. $?$ is associative; if $i,j,k\in G$ then $\left(i?j\right)?k=i?\left(j?k\right)$, and
2. $G$ 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 an operation $$\begin{array}{rcl}+:G\times G& ⟶& G\\ \left(i,j\right)& ⟼& i+j\end{array}$$ such that

1. $G$ is a monoid, and
2. if $i,j\in G$ then $i+j=j+i$.

A group is a set $G$ with an operation $$\begin{array}{rcl}?:G\times G& ⟶& G\\ \left(i,j\right)& ⟼& i?j\end{array}$$ such that

1. $?$ is associative; if $i,j,k\in G$ then $\left(i?j\right)?k=i?\left(j?k\right)$,
2. $G$ has an identity; there exists an element $!\in G$ such that if $y\in G$ then $!?y=y?!=y$, and
3. $G$ has inverses; if $y\in G$ there is an element $y^{\#}\in G$ such that $y?y^{\#}=y^{\#}?y=!$ where $!$ is the identity in $G$.

A abelian group is a set $G$ with an operation $$\begin{array}{rcl}+:G\times G& ⟶& G\\ \left(i,j\right)& ⟼& i+j\end{array}$$ such that

1. $G$ is a group, and
2. if $i,j\in G$ then $i+j=j+i$.

A ring without identity is a set $R$ with two operations $$\begin{array}{rcl}+:R\times R& ⟶& R\\ \left(i,j\right)& ⟼& i+j\end{array}\text{and}\begin{array}{rcl}\times :R\times R& ⟶& R\\ \left(i,j\right)& ⟼& i\times j=ij\end{array}$$ such that

1. $R$ with the operation $+$ is an abelian group,
2. $R$ with the operation $\times$ is a monoid without identity, and
3. $R$ has distributive laws; if $i,j,k\in R$ then $i\left(j+k\right)=ij+ik$ and $\left(i+j\right)k=ik+jk$.

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 the operation $+$) 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 the operation $+$) then there is an element $y^{-1}\in 𝔻$ with $y{y}^{-1}=y^{-1}y=1$.

The integers $\mathbb{Z}$ with the addition operation is an abelian group. The integers $\mathbb{Z}$ with the addition and multiplication operations is a ring. The rationals $\mathbb{Q}$ with the operations addition and multiplication is a field.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)