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.
The ring is named after Nathan Jacobson.
Famous quotes containing the word ring:
“I was exceedingly interested by this phenomenon, and already felt paid for my journey. It could hardly have thrilled me more if it had taken the form of letters, or of the human face. If I had met with this ring of light while groping in this forest alone, away from any fire, I should have been still more surprised. I little thought that there was such a light shining in the darkness of the wilderness for me.”
—Henry David Thoreau (1817–1862)