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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 , and ) 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