Noetherian rings and are key concepts in commutative algebra. They provide a foundation for understanding structure and polynomial rings, which are crucial in algebraic geometry.
These ideas help us grasp how ideals behave in rings and how this behavior extends to polynomial rings. Hilbert's basis theorem, in particular, shows that polynomial rings over Noetherian rings are also Noetherian, a powerful result with wide-ranging applications.
Hilbert's Basis Theorem
Statement and Proof
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Hilbert's basis theorem states that if R is a , then the R[x] is also Noetherian
The proof of Hilbert's basis theorem relies on the following key concepts:
Monomial ordering: A total order on the monomials in a polynomial ring that is compatible with multiplication and has the property that 1 is the smallest monomial
Division algorithm for polynomials: Given polynomials f and g, there exist unique polynomials q and r such that f=qg+r, where r has a smaller degree than g or r=0
The proof proceeds by contradiction:
Assume that R[x] is not Noetherian
Construct an infinite strictly increasing chain of ideals in R
This contradicts the assumption that R is Noetherian, proving the theorem
Monomial Orderings and Division Algorithm
A monomial ordering is essential for the division algorithm and Gröbner basis theory
Examples of monomial orderings include lexicographic order, graded lexicographic order, and graded reverse lexicographic order
The division algorithm for polynomials is a generalization of the Euclidean division algorithm for integers
It allows for the reduction of a polynomial by a set of polynomials, which is crucial in the computation of Gröbner bases
The existence and uniqueness of the quotient and remainder in the division algorithm rely on the properties of the monomial ordering
Noetherian Rings
Definition and Examples
A ring R is Noetherian if it satisfies the (ACC) on ideals
ACC states that every ascending chain of ideals I1⊆I2⊆I3⊆… eventually stabilizes, i.e., there exists an n such that In=In+1=In+2=…
Examples of Noetherian rings:
Fields (Q, R, C)
Principal ideal domains (PIDs) (Z, k[x] for a k)
Finitely generated algebras over a field (k[x1,…,xn] for a field k)
Examples of non-Noetherian rings:
The ring of polynomials in infinitely many variables k[x1,x2,…] over a field k
The ring of continuous functions on the real line C(R)
The ring of formal power series k[[x]] over a field k
Finitely Generated Ideals
An ideal I in a ring R is finitely generated if there exist elements a1,…,an in I such that every element of I can be written as a linear combination of a1,…,an with coefficients in R
In other words, I=(a1,…,an)={r1a1+…+rnan∣r1,…,rn∈R}
Finitely generated ideals are essential in the study of Noetherian rings, as they provide a way to characterize these rings
Characterization of Noetherian Rings
Equivalence with Finitely Generated Ideals
A ring R is Noetherian if and only if every ideal in R is finitely generated
To prove the forward direction (R Noetherian ⟹ every ideal is finitely generated):
Assume R is Noetherian and let I be an ideal in R
Consider the set S of all finitely generated ideals contained in I
The ACC implies that S has a maximal element J, which must be equal to I, proving that I is finitely generated
To prove the reverse direction (every ideal is finitely generated ⟹R Noetherian):
Assume every ideal in R is finitely generated
Given an ascending chain of ideals I1⊆I2⊆I3⊆…, consider the ideal I=⋃nIn
By assumption, I is finitely generated, so I=In for some n, implying that the chain stabilizes
Maximal Ideals and Prime Ideals
In a Noetherian ring, every ideal is finitely generated, which has important consequences for the structure of the ring
Every Noetherian ring has a finite number of maximal ideals
A maximal ideal is an ideal M that is not contained in any other proper ideal
Every Noetherian ring satisfies the descending chain condition (DCC) on prime ideals
A prime ideal is an ideal P such that if ab∈P, then either a∈P or b∈P
DCC states that every descending chain of prime ideals P1⊇P2⊇P3⊇… eventually stabilizes
Applications of Hilbert's Basis Theorem
Proving Rings are Noetherian
Hilbert's basis theorem can be used to prove that certain rings are Noetherian
For example, polynomial rings over Noetherian rings, such as Z[x] and R[x,y], are Noetherian
The theorem can be applied to show that finitely generated modules over a Noetherian ring are Noetherian
A M over a ring R is Noetherian if it satisfies the ACC on submodules
If R is Noetherian and M is finitely generated, then M is Noetherian
Hilbert's Nullstellensatz and Primary Decomposition
Hilbert's basis theorem is a key tool in proving the Hilbert's Nullstellensatz
The Nullstellensatz relates ideals in polynomial rings to algebraic sets in affine space
It states that if k is an algebraically closed field and I is an ideal in k[x1,…,xn], then the radical of I is equal to the ideal of all polynomials vanishing on the algebraic set defined by I
The theorem can be used to prove that certain rings have primary decomposition
A primary decomposition of an ideal I is a representation of I as a finite intersection of primary ideals
In a Noetherian ring, every ideal has a primary decomposition
Invariant Theory
Hilbert's basis theorem is essential in the study of invariant theory
Invariant theory is concerned with the study of polynomials that are invariant under the action of a group
The theorem guarantees the existence of finite generating sets for rings of invariants under group actions
For example, the ring of invariants of the symmetric group acting on a polynomial ring is finitely generated
of invariant rings has important consequences in algebraic geometry and representation theory