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Rings: definition and basic properties

We now give an introduction to another type of algebraic structure, called ring. The exposition here will be faster. The emphasis is on the definition of a ring and a field, understanding the few basic examples, and realising that the concepts of a subring, homomorphism and quotient can be defined in a similar way to groups.


\begin{de}
A set $R$\ with two operations $+$\ and $\cdot$\ defined on it
is a {...
...displaymath}
x(y+z)=xy+xz,\ (x+y)z=xz+yz.
\end{displaymath}\end{itemize}\end{de}

We will see a number of examples of rings in the next section. First we are going to list some basic consequences of the axioms. We note that many basic properties involving $+$ (such as $x+0=0+x=x$, $x+(-x)=0$ and $-(-x)=x$) follow from the definition of a group and Theorem 2.8.


\begin{thm}
The following hold in any ring $R$:
\begin{itemize}
\item[\rm (i)]
$...
...-x)y=x(-y)=-(xy)$\ and $(-x)(-y)=xy$\ for all $x,y\in R$.
\end{itemize}\end{thm}


\begin{proof}
(i)
$x0=x(0+0)=x0+x0\Rightarrow x0=0$.
\par (ii)
$xy+(-x)y=(x+(-x))y=0y=0\Rightarrow (-x)y=-(xy)$;
$(-x)(-y)=-x(-y)=-(-(xy))=xy$.
\end{proof}


\begin{exc}
Prove that for all $x,y,z,t\in R$\ we have
\begin{displaymath}
(x+y)(z+t)=xz+xt+yz+yt.
\end{displaymath}\end{exc}



Edmund F Robertson

11 September 2006