= Ideals of subrings of the rational numbers
{title2=$\mathbb Z\subseteq R\subseteq\mathbb Q\implies I=(I\cap\mathbb Z)R$}
Every <subring> $R$ of the <rational numbers> containing $\mathbb Z$ is a <principal ideal domain>. If $a/b\in R$ is reduced, <Bézout's identity> gives integers $u,v$ with $ua+vb=1$, hence $1/b=u(a/b)+v\in R$. Thus every reduced denominator occurring in $R$ is a <unit>.
For an <ideal> $I$, write $I\cap\mathbb Z=d\mathbb Z$. Then $dR\subseteq I$. If $a/b\in I$ is reduced, $a=b(a/b)\in I\cap\mathbb Z$, so $a=dc$ for some integer $c$. Since $1/b\in R$, $a/b=d(c/b)\in dR$. Therefore $I=dR$, including the zero ideal. This is a concrete instance of extending and contracting <ideals> under <localization of a ring>.
Back to article page
