13B21-Integral.tex

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Let $B$ be a ring with a subring $A$. We will assume that $A$ is contained in the center of $B$ (in particular, $A$ is commutative). An element $x \in B$ is \emph{integral} over $A$ if there exist elements $a_0, \dots, a_{n-1} \in A$ such that
$$
x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0.
$$
The ring $B$ is \emph{integral} over $A$ if every element of $B$ is integral over $A$.
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