2.2 Noetherian rings and Hilbert's basis theorem

Noetherian rings and Hilbert's basis theorem are key concepts in commutative algebra. They provide a foundation for understanding ideal 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

• Hilbert's basis theorem states that if $R$ is a Noetherian ring, then the polynomial ring $R[x]$ 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 ascending chain condition (ACC) on ideals
  • ACC states that every ascending chain of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq \ldots$ eventually stabilizes, i.e., there exists an $n$ such that $I_n = I_{n+1} = I_{n+2} = \ldots$
• Examples of Noetherian rings:
  • Fields ($\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$)
  • Principal ideal domains (PIDs) ($\mathbb{Z}$, $k[x]$ for a field $k$)
  • Finitely generated algebras over a field ($k[x_1, \ldots, x_n]$ for a field $k$)
• Examples of non-Noetherian rings:
  • The ring of polynomials in infinitely many variables $k[x_1, x_2, \ldots]$ over a field $k$
  • The ring of continuous functions on the real line $C(\mathbb{R})$
  • 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 $a_1, \ldots, a_n$ in $I$ such that every element of $I$ can be written as a linear combination of $a_1, \ldots, a_n$ with coefficients in $R$
  • In other words, $I = (a_1, \ldots, a_n) = \{r_1a_1 + \ldots + r_na_n \mid r_1, \ldots, r_n \in R\}$
• 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 $\implies$ 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 $\implies$ $R$ Noetherian):
  • Assume every ideal in $R$ is finitely generated
  • Given an ascending chain of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq \ldots$, consider the ideal $I = \bigcup_n I_n$
  • By assumption, $I$ is finitely generated, so $I = I_n$ 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 $ab \in P$, then either $a \in P$ or $b \in P$
  • DCC states that every descending chain of prime ideals $P_1 \supseteq P_2 \supseteq P_3 \supseteq \ldots$ 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 $\mathbb{Z}[x]$ and $\mathbb{R}[x, y]$, are Noetherian
• The theorem can be applied to show that finitely generated modules over a Noetherian ring are Noetherian
  • A module $M$ 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 $k[x_1, \ldots, x_n]$, 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
• Finite generation of invariant rings has important consequences in algebraic geometry and representation theory