The Stacks project

Lemma 10.36.5. Let $R \to S$ be a ring map. The following are equivalent

  1. $R \to S$ is finite,

  2. $R \to S$ is integral and of finite type, and

  3. there exist $x_1, \ldots , x_ n \in S$ which generate $S$ as an algebra over $R$ such that each $x_ i$ is integral over $R$.

Proof. Clear from Lemma 10.36.4. $\square$

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