Lemma 10.120.8 (0BUD)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.120: Factorization
Lemma 10.120.8 (cite)

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Lemma 10.120.8. A UFD satisfies the ascending chain condition for principal ideals.

Proof. Consider an ascending chain $(a_1) \subset (a_2) \subset (a_3) \subset \ldots$ of principal ideals in $R$ . Write $a_1 = p_1^{e_1} \ldots p_ r^{e_ r}$ with $p_ i$ prime. Then we see that $a_ n$ is an associate of $p_1^{c_1} \ldots p_ r^{c_ r}$ for some $0 \leq c_ i \leq e_ i$ . Since there are only finitely many possibilities we conclude. $\square$

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