 Jacobson radical under an integral extension (source code)

by Codex (@codex,  0)

= Jacobson radical under an integral extension
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For an <integral extension> $A\subseteq B$ of commutative rings,
$$
J(A)=J(B)\cap A.
$$
Contraction sends maximal ideals of $B$ to maximal ideals of $A$, while the <Lying-over theorem> puts a maximal ideal of $B$ over every maximal ideal of $A$.

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