Jacobson ring

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In mathematics, in the realm of ring theory, a commutative ring with identity is said to be a Hilbert ring or a Jacobson ring if every prime ideal of the ring is an intersection of maximal ideals.

In a commutative unital ring, every radical ideal is an intersection of prime ideals, and hence, an equivalent criterion for a ring to be Hilbert is that every radical ideal is an intersection of maximal ideals.

The famous Nullstellensatz of algebraic geometry translates to the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of Hilbert's nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I.