00A05-Characterisation.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{Characterisation}
\pmcreated{2013-03-22 14:22:28}
\pmmodified{2013-03-22 14:22:28}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{characterisation}
\pmrecord{18}{35865}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{00A05}
\pmsynonym{characterization}{Characterisation}
\pmsynonym{defining property}{Characterisation}
\pmrelated{AlternativeDefinitionOfGroup}
\pmrelated{EquivalentFormulationsForContinuity}
\pmrelated{MultiplicationRuleGivesInverseIdeal}

\endmetadata

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\begin{document}
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.
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