next up previous contents
Next: Subrings, ideals and quotients Up: MT2002 Algebra Previous: Examples of rings   Contents

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