Definition 10.35.1. Let $\varphi : R \to S$ be a ring map.

An element $s \in S$ is

*integral over $R$*if there exists a monic polynomial $P(x) \in R[x]$ such that $P^\varphi (s) = 0$, where $P^\varphi (x) \in S[x]$ is the image of $P$ under $\varphi : R[x] \to S[x]$.The ring map $\varphi $ is

*integral*if every $s \in S$ is integral over $R$.

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