Jacobson ring
Encyclopedia
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 ideal
s.
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the realm of ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with identity is said to be a Hilbert ring or a Jacobson ring if every prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
of the ring is an intersection of maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
s.
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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
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
Nathan Jacobson
Nathan Jacobson was an American mathematician....
.