The Stacks project

Lemma 10.120.2. Let $R$ be a domain. Let $x, y \in R$. Then $x$, $y$ are associates if and only if $(x) = (y)$.

Proof. If $x = uy$ for some unit $u \in R$, then $(x) \subset (y)$ and $y = u^{-1}x$ so also $(y) \subset (x)$. Conversely, suppose that $(x) = (y)$. Then $x = fy$ and $y = gx$ for some $f, g \in A$. Then $x = fg x$ and since $R$ is a domain $fg = 1$. Thus $x$ and $y$ are associates. $\square$

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