Definition 10.38.1. Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ be an ideal. We say an element $g \in S$ is integral over $I$ if there exists a monic polynomial $P = x^ d + \sum _{j < d} a_ j x^ j$ with coefficients $a_ j \in I^{d-j}$ such that $P^\varphi (g) = 0$ in $S$.

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