Integral extension

Remarks

Example: $\Z \subseteq \Z[i]$ is integral. The element $a + bi$ with $a, b \in \Z$ is a root of the monic polynomial $x^2 - 2a x + a^2 + b^2$ with integer coefficients.

Non-example: $\Z \subseteq \Q$ is not integral, because $1/2$ is not integral over $\Z$.

An extension is integral and of finite type if and only if it is finite. Integrality is transitive, and it is kept by quotients and localizations. If $B$ is integral over $A$, then $B/J$ is integral over $A/(J \cap A)$ for every ideal $J$ of $B$. Also $S^{-1}B$ is integral over $S^{-1}A$ for every multiplicative set $S$ of $A$.