Jacobson ring

Summary

A Jacobson ring, also called a Hilbert ring, is a commutative ring in which every prime ideal is an intersection of maximal ideals. In such a ring, the maximal ideals alone determine all radical ideals, as in Hilbert's Nullstellensatz. Fields, $\Z$, and every finitely generated algebra over a Jacobson ring are Jacobson rings; this last fact is a general form of the Nullstellensatz. In geometry, the closed points of the spectrum of a Jacobson ring are dense in every closed subset.