# The principal fractional ideals of an integral domain form a directed partially ordered group

Let $R$ be an integral domain and $K$ be its quotient field. Let $G = \{aR: a\in K^{\times}\}$. Then $G$ is a partially ordered group under $aR\leq bR$ iff $bR\subseteq aR$.

But I have hard time to show $G$ is a directed partially ordered group.

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Let $a,a'\in K^{\times}$. Write $a=u/v$ and $a'=u'/v'$ with $u,v,u',v'\in R$, $v\neq 0$, $v'\neq 0$. Since $uu'=a(vu')$ and $uu'=a'(v'u)$, then $aR\le uu'R$ and $a'R\le uu'R$.