HalfFactorialRing 2008-10-27 14:53:06 2008-10-27 18:07:17 Definition UFD HFD % 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} An integral domain $D$ is called a {\em half-factorial ring} (HFD) if it satisfies the following conditions: \begin{itemize} \item Every nonzero element of $D$ that is not a unit can be factored into a product of a finite number of irreducibles. \item If\, $p_1p_2\cdots p_m$\, and\, $q_1q_2\cdots q_n$\, are two factorizations of the same element $a$ into irreducibles, then\, $m = n$. \end{itemize} If, in \PMlinkescapetext{addition}, the irreducibles $p_i$ and $q_j$ are always pairwise associates, then $D$ is a factorial ring (UFD).\\ For example, many \PMlinkname{orders}{OrderInAnAlgebra} in the maximal order of an algebraic number field are half-factorial rings, e.g. $\mathbb{Z}[3\sqrt{2}]$ is a HFD but not a UFD (see \PMlinkexternal{this paper}{http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdf}).