characterisation Characterization 2004-05-21 09:03:46 2007-10-24 15:50:14 Definition property % 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 In mathematics, {\em characterisation} usually means a property or a condition to define a certain notion. \,A notion may, under some presumptions, have different \PMlinkescapetext{equivalent} ways to define it. For example, let $R$ be a commutative ring with non-zero unity (the presumption). \,Then the following are equivalent: (1) All finitely generated regular ideals of $R$ are invertible. (2) The \PMlinkescapetext{formula} \,$(a,\,b)(c,\,d) = (ac,\,bd,\,(a+b)(c+d))$\, for multiplying ideals of $R$ is valid always when at least one of the elements $a$, $b$, $c$, $d$ of $R$ is not zero-divisor. (3) Every overring of $R$ is integrally closed. Each of these conditions is sufficient (and necessary) for characterising and defining the Pr\"ufer ring.