group of unitsGroupOfUnits2004-10-05 16:56:162008-03-30 04:05:37Theoremunit% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}\begin{thmplain}
\, The set $E$ of units of a ring $R$ forms a group with respect to ring multiplication.
\end{thmplain}
{\em Proof.}\, If $u$ and $v$ are two units, then there are the elements $r$ and $s$ of $R$ such that\, $ru = ur = 1$\, and\, $sv = vs = 1$.\, Then we get that\,$(sr)(uv) = s(r(uv)) = s((ru)v) = s(1v) = sv = 1$,\, similarly\, $(uv)(sr) = 1$.\, Thus also $uv$ is a unit, which means that $E$ is closed under multiplication.\, Because\, $1 \in E$\, and along with $u$ also its inverse $r$ belongs to $E$, the set $E$ is a group.
\textbf{Corollary.}\, In a commutative ring, a ring product is a unit iff all \PMlinkescapetext{factors} are units.
\textbf{Examples}
\begin{enumerate}
\item When\, $R = \mathbb{Z}$, then\, $E = \{1,\,-1\}$.
\item When\, $R = \mathbb{Z}[i]$,\, the ring of Gaussian integers, then\,
$E = \{1,\,i,\,-1,\,-i\}$.
\item When\, $R = \mathbb{Z}[\sqrt{3}]$, \PMlinkname{then}{UnitsOfQuadraticFields}\,
$E = \{\pm(2\!+\!\sqrt{3})^n\,\vdots\,\,\, n\in\mathbb{Z}\}$.
\item When\, $R = K[X]$\, where $K$ is a field, then\, $E = K\!\smallsetminus\!\{0\}$.
\item When\,
$R = \{0\!+\!\mathbb{Z},\,1\!+\!\mathbb{Z},\,\ldots,\,
m\!-\!1\!+\!\mathbb{Z}\}$\, is the residue class ring modulo $m$, then\, $E$ consists of the prime classes modulo $m$, i.e. the residue classes $l\!+\!\mathbb{Z}$ satisfying\, $\gcd(l, m) = 1$.
\end{enumerate}