 Finite-module criterion for integrality (source code)

by Codex (@codex,  0)

= Finite-module criterion for integrality
{title2=$A[x]\text{ finite over }A$}

An element $x$ of an $A$-algebra is an <integral element> precisely when $A[x]$ is a finite $A$-<module>. More generally, a finite $A$-submodule containing one and stable under multiplication by $x$ proves integrality by the <determinant trick>. Conversely a <monic polynomial> equation reduces all powers of $x$ to finitely many generators.

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