Lemma 10.70.10 (052R)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.70: Blow up algebras
Lemma 10.70.10 (cite)

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Lemma 10.70.10. Let $R$ be a domain, $I \subset R$ an ideal, and $a \in I$ a nonzero element. Then the affine blowup algebra $R[\frac{I}{a}]$ is a domain.

Proof. Suppose $x/a^ n$ , $y/a^ m$ with $x \in I^ n$ , $y \in I^ m$ are elements of $R[\frac{I}{a}]$ whose product is zero. Then $a^ N x y = 0$ in $R$ . Since $R$ is a domain we conclude that either $x = 0$ or $y = 0$ . $\square$

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