A commutative ring is Jacobson when every prime ideal is the intersection of the maximal ideals containing it. Equivalently, for every ideal , its radical is the intersection of the maximal ideals containing .
For a finite-type integer algebra, a nonnilpotent element gives a nonzero localization of a ring . A maximal ideal there has finite residue field by the finite-field theorem for finitely generated integer algebras. The image of in that field is a finite integral domain, hence a field. Its kernel is therefore maximal in and avoids . Nilpotents lie in every maximal ideal, proving the radical equality. Applying the same argument to every prime quotient shows that these algebras are Jacobson rings. Merely contracting a localized maximal ideal without the finite-image argument would not prove maximality.
Every ring integral over a Jacobson ring is Jacobson. After quotienting by a prime, an integral equation for a nonzero element has nonzero constant term; choose a maximal ideal of the base avoiding that term and apply the Lying-over theorem.
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In the context of commutative algebra, a Jacobson ring is a ring that satisfies certain properties related to its prime ideals and maximal ideals. Specifically, a ring \( R \) is called a **Jacobson ring** if the intersection of all maximal ideals of \( R \) is equal to the nilradical of \( R \).