Lemma 10.25.3 (02LW)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.25: Zerodivisors and total rings of fractions
Lemma 10.25.3 (cite)

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Lemma 10.25.3. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset consisting of nonzerodivisors. Then $Q(R) \cong Q(S^{-1}R)$. In particular $Q(R) \cong Q(Q(R))$.

Proof. If $x \in S^{-1}R$ is a nonzerodivisor, and $x = r/f$ for some $r \in R$, $f \in S$, then $r$ is a nonzerodivisor in $R$. Whence the lemma. $\square$

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