next up previous contents
Next: Special types of rings Up: MT2002 Algebra Previous: Rings: definition and basic   Contents

Examples of rings


\begin{ex}
Each of the number sets $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$\ and...
...b{C}$forms a ring with respect to ordinary addition and multiplication.
\end{ex}


\begin{exc}
For every $m\in {\mathbb{Z}}$\ the set $m{\mathbb{Z}}=\{ ma\st a\in{...
...}\}$forms a ring with respect to ordinary addition and multiplication.
\end{exc}


\begin{ex}
The set ${\mathbb{Z}}_n$\ is a ring with respect to addition and multiplication
modulo $n$.
\end{ex}


\begin{ex}
A (real) polynomial is a formal expression of the form
\begin{display...
...an in a similar way construct
the ring $R[x]$\ of polynomials over $x$.
\end{ex}


\begin{ex}
The set $M_n({\mathbb{R}})$\ of all $n\times n$\ matrices
with entrie...
...
the ring $M_n(R)$\ of all $n\times n$\ matrices with entries
from $R$.
\end{ex}


\begin{exc}
The set
\begin{displaymath}
\left\{ \left( \begin{array}{cc}0&a\\ 0&...
... ring with respect to the ordinary matrix addition and multiplication.
\end{exc}


\begin{ex}
% latex2html id marker 4659The set
\begin{displaymath}
{\mathbb{H}}...
...9i+9+6k-6j\\ &&-3j-3k+2-2i+15k-15j-10i-10=
7-9i-20j+22k.
\end{eqnarray*}\end{ex}


\begin{ex}
Let $G$\ be any abelian group, written additively. On $g$\ define a m...
...or all $x,y\in G$. This makes $G$\ into a
(not very interesting!) ring.
\end{ex}



Edmund F Robertson

11 September 2006