Hilbert's basis theorem

Short description: Polynomial rings are Noetherian rings

In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

Statement

If [math]\displaystyle{ R }[/math] is a ring, let [math]\displaystyle{ R[X] }[/math] denote the ring of polynomials in the indeterminate [math]\displaystyle{ X }[/math] over [math]\displaystyle{ R }[/math]. Hilbert proved that if [math]\displaystyle{ R }[/math] is "not too large", in the sense that if [math]\displaystyle{ R }[/math] is Noetherian, the same must be true for [math]\displaystyle{ R[X] }[/math]. Formally,

Hilbert's Basis Theorem. If [math]\displaystyle{ R }[/math] is a Noetherian ring, then [math]\displaystyle{ R[X] }[/math] is a Noetherian ring.

Corollary. If [math]\displaystyle{ R }[/math] is a Noetherian ring, then [math]\displaystyle{ R[X_1,\dotsc,X_n] }[/math] is a Noetherian ring.

This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. Hilbert proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.

Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.

Proof

Theorem. If [math]\displaystyle{ R }[/math] is a left (resp. right) Noetherian ring, then the polynomial ring [math]\displaystyle{ R[X] }[/math] is also a left (resp. right) Noetherian ring.

Remark. We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.

First proof

Suppose [math]\displaystyle{ \mathfrak a \subseteq R[X] }[/math] is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials [math]\displaystyle{ \{ f_0, f_1, \ldots \} }[/math] such that if [math]\displaystyle{ \mathfrak b_n }[/math] is the left ideal generated by [math]\displaystyle{ f_0, \ldots, f_{n-1} }[/math] then [math]\displaystyle{ f_n \in \mathfrak a \setminus \mathfrak b_n }[/math] is of minimal degree. By construction, [math]\displaystyle{ \{\deg(f_0), \deg(f_1), \ldots \} }[/math] is a non-decreasing sequence of natural numbers. Let [math]\displaystyle{ a_n }[/math] be the leading coefficient of [math]\displaystyle{ f_n }[/math] and let [math]\displaystyle{ \mathfrak{b} }[/math] be the left ideal in [math]\displaystyle{ R }[/math] generated by [math]\displaystyle{ a_0,a_1,\ldots }[/math]. Since [math]\displaystyle{ R }[/math] is Noetherian the chain of ideals

[math]\displaystyle{ (a_0)\subset(a_0,a_1)\subset(a_0,a_1,a_2) \subset \cdots }[/math]

must terminate. Thus [math]\displaystyle{ \mathfrak b = (a_0,\ldots ,a_{N-1}) }[/math] for some integer [math]\displaystyle{ N }[/math]. So in particular,

[math]\displaystyle{ a_N=\sum_{i\lt N} u_{i}a_{i}, \qquad u_i \in R. }[/math]

Now consider

[math]\displaystyle{ g = \sum_{i\lt N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i}, }[/math]

whose leading term is equal to that of [math]\displaystyle{ f_N }[/math]; moreover, [math]\displaystyle{ g\in\mathfrak b_N }[/math]. However, [math]\displaystyle{ f_N \notin \mathfrak b_N }[/math], which means that [math]\displaystyle{ f_N - g \in \mathfrak a \setminus \mathfrak b_N }[/math] has degree less than [math]\displaystyle{ f_N }[/math], contradicting the minimality.

Second proof

Let [math]\displaystyle{ \mathfrak a \subseteq R[X] }[/math] be a left ideal. Let [math]\displaystyle{ \mathfrak b }[/math] be the set of leading coefficients of members of [math]\displaystyle{ \mathfrak a }[/math]. This is obviously a left ideal over [math]\displaystyle{ R }[/math], and so is finitely generated by the leading coefficients of finitely many members of [math]\displaystyle{ \mathfrak a }[/math]; say [math]\displaystyle{ f_0, \ldots, f_{N-1} }[/math]. Let [math]\displaystyle{ d }[/math] be the maximum of the set [math]\displaystyle{ \{\deg(f_0),\ldots, \deg(f_{N-1})\} }[/math], and let [math]\displaystyle{ \mathfrak b_k }[/math] be the set of leading coefficients of members of [math]\displaystyle{ \mathfrak a }[/math], whose degree is [math]\displaystyle{ \le k }[/math]. As before, the [math]\displaystyle{ \mathfrak b_k }[/math] are left ideals over [math]\displaystyle{ R }[/math], and so are finitely generated by the leading coefficients of finitely many members of [math]\displaystyle{ \mathfrak a }[/math], say

[math]\displaystyle{ f^{(k)}_{0}, \ldots, f^{(k)}_{N^{(k)}-1} }[/math]

with degrees [math]\displaystyle{ \le k }[/math]. Now let [math]\displaystyle{ \mathfrak a^*\subseteq R[X] }[/math] be the left ideal generated by:

[math]\displaystyle{ \left\{f_{i},f^{(k)}_{j} \, : \ i\lt N,\, j\lt N^{(k)},\, k\lt d \right\}\!\!\;. }[/math]

We have [math]\displaystyle{ \mathfrak a^*\subseteq\mathfrak a }[/math] and claim also [math]\displaystyle{ \mathfrak a\subseteq\mathfrak a^* }[/math]. Suppose for the sake of contradiction this is not so. Then let [math]\displaystyle{ h\in \mathfrak a \setminus \mathfrak a^* }[/math] be of minimal degree, and denote its leading coefficient by [math]\displaystyle{ a }[/math].

Case 1: [math]\displaystyle{ \deg(h)\ge d }[/math]. Regardless of this condition, we have [math]\displaystyle{ a\in \mathfrak b }[/math], so [math]\displaystyle{ a }[/math] is a left linear combination

[math]\displaystyle{ a=\sum_j u_j a_j }[/math]

of the coefficients of the [math]\displaystyle{ f_j }[/math]. Consider

[math]\displaystyle{ h_0 =\sum_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j}, }[/math]

which has the same leading term as [math]\displaystyle{ h }[/math]; moreover [math]\displaystyle{ h_0 \in \mathfrak a^* }[/math] while [math]\displaystyle{ h\notin\mathfrak a^* }[/math]. Therefore [math]\displaystyle{ h - h_0 \in \mathfrak a\setminus\mathfrak a^* }[/math] and [math]\displaystyle{ \deg(h - h_0) \lt \deg(h) }[/math], which contradicts minimality.

Case 2: [math]\displaystyle{ \deg(h) = k \lt d }[/math]. Then [math]\displaystyle{ a\in\mathfrak b_k }[/math] so [math]\displaystyle{ a }[/math] is a left linear combination

[math]\displaystyle{ a=\sum_j u_j a^{(k)}_j }[/math]

of the leading coefficients of the [math]\displaystyle{ f^{(k)}_j }[/math]. Considering

[math]\displaystyle{ h_0=\sum_j u_j X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j}, }[/math]

we yield a similar contradiction as in Case 1.

Thus our claim holds, and [math]\displaystyle{ \mathfrak a = \mathfrak a^* }[/math] which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of [math]\displaystyle{ X }[/math] multiplying the factors were non-negative in the constructions.

Applications

Let [math]\displaystyle{ R }[/math] be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries.

1. By induction we see that [math]\displaystyle{ R[X_0,\dotsc,X_{n-1}] }[/math] will also be Noetherian.
2. Since any affine variety over [math]\displaystyle{ R^n }[/math] (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal [math]\displaystyle{ \mathfrak a\subset R[X_0, \dotsc, X_{n-1}] }[/math] and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces.
3. If [math]\displaystyle{ A }[/math] is a finitely-generated [math]\displaystyle{ R }[/math]-algebra, then we know that [math]\displaystyle{ A \simeq R[X_0, \dotsc, X_{n-1}] / \mathfrak a }[/math], where [math]\displaystyle{ \mathfrak a }[/math] is an ideal. The basis theorem implies that [math]\displaystyle{ \mathfrak a }[/math] must be finitely generated, say [math]\displaystyle{ \mathfrak a = (p_0,\dotsc, p_{N-1}) }[/math], i.e. [math]\displaystyle{ A }[/math] is finitely presented.

Formal proofs

Formal proofs of Hilbert's basis theorem have been verified through the Mizar project (see HILBASIS file) and Lean (see ring_theory.polynomial).

Further reading

• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.

• {{citation | last=Roman | first=Stephen