unit Unit 2001-11-04 23:02:58 2006-05-30 15:37:10 Definition algebraic unit Ring Factorization \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} Let $R$ be a ring with multiplicative identity $1$. We say that $u\in R$ is an unit (or unital) if $u$ divides $1$ (denoted $u \mid 1$). That is, there exists an $r\in R$ such that $1=ur=ru$. Notice that $r$ will be the multiplicative inverse (in the ring) of $u$, so we can characterize the units as those elements of the ring having multiplicative inverses. In the special case that $R$ is the ring of integers of an algebraic number field $K$, the units of $R$ are sometimes called the {\em algebraic units} of $K$ (and also the units of $K$).\, They are determined by Dirichlet's unit theorem.