Lemma 10.120.2 (034Q)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.120: Factorization
Lemma 10.120.2 (cite)

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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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