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Special types of rings

$\begin{de} A ring $R$\ is said to be {\em commutative} if the multiplication in $R$\ is commutative, i.e. if $xy=yx$\ holds for all $x,y\in R$. \end{de}$

$\begin{exs} The rings $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, ${... ...e. The rings $\mathbb{H}$\ and $M_n(\mathbb{R})$\ are not commutative. \end{exs}$

$\begin{de} A ring $R$\ is said to be a {\em ring with identity} if there is a ne... ...ted by $1$) for multiplication; thus $1x=x1=x$\ holds for all $x\in R$. \end{de}$

$\begin{exs} % latex2html id marker 4680All the rings mentioned in Examples \re... ...rom Exercises \ref{exc73} and \ref{exc74} are not rings with identity. \end{exs}$

$\begin{de} A ring $R$\ is said to be an {\em integral domain} if it is commutative and $xy=0$\ implies $x=0$\ or $y=0$for all $x,y\in R$. \end{de}$

$\begin{exs} All $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$\ and $\mathbb{C}$\ are ... ...ly if $n$\ is a prime. $M_n({\mathbb{R}})$\ is not an integral domain. \end{exs}$

$\begin{de} A ring $R$\ is a {\em division ring} if $R\setminus\{0\}$\ forms a gr... ...entity and every non-zero element of $R$\ has a multiplicative inverse. \end{de}$

$\begin{de} A ring $R$\ is a {\em field} if $R\setminus \{0\}$\ forms an abelian group, i.e. if $R$\ is a commutative division ring. \end{de}$

$\begin{exs} $\mathbb{Q}$, $\mathbb{R}$\ and $\mathbb{C}$\ are fields. $\mathbb{H... ...$, $M_n({\mathbb{R}})$\ and ${\mathbb{R}}[x]$\ are not division rings. \end{exs}$

$\begin{ex} % latex2html id marker 4705${\mathbb{Z}}_n$\ is a field if and only... ...{\mathbb{Z}}_n\setminus\{0\}$is a group if and only if $n$\ is a prime. \end{ex}$

Edmund F Robertson

11 September 2006