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Hilbert's basis theorem

Hilbert's basis theorem is a foundational result in stating that if R is a , then the R in one indeterminate over R is also Noetherian, implying that every ideal in R is finitely generated. This theorem was originally proved by in 1890 as part of his work on the theory of algebraic forms and invariants. The theorem extends inductively to polynomial rings in any finite number of indeterminates, establishing that rings like k[x_1, \dots, x_n] over a k are Noetherian. Its proof relies on the ascending chain condition for s, showing that any in R has a finite basis by considering leading terms with respect to a suitable ordering. Hilbert's non-constructive approach marked a shift toward abstract methods in algebra, influencing the development of modern . Key applications include the ideal-variety correspondence in , where finitely generated ideals correspond to algebraic varieties, and the foundations of theory for computational algebra. The result also underpins and Nullstellensatz, facilitating the study of solutions to systems. Generalizations extend to rings and modules over Noetherian rings, broadening its scope in .

Background Concepts

Polynomial Rings

A polynomial ring R[x_1, \dots, x_n] over a commutative ring R with unity is formed by the set of all finite formal sums f = \sum_{\alpha} a_{\alpha} x^{\alpha}, where \alpha = (\alpha_1, \dots, \alpha_n) is a multi-index of non-negative integers, a_{\alpha} \in R are coefficients, and x^{\alpha} = x_1^{\alpha_1} \cdots x_n^{\alpha_n} are monomials in the indeterminates x_1, \dots, x_n. Addition is performed termwise by adding corresponding coefficients, and multiplication is defined by the rule that monomials multiply via x^\alpha \cdot x^\beta = x^{\alpha + \beta} (componentwise exponent addition) extended by distributivity over addition, making R[x_1, \dots, x_n] a commutative ring with unity whenever R is commutative with unity. Key properties include the total degree of f, defined as \deg(f) = \max \{ |\alpha| : a_{\alpha} \neq 0 \}, where |\alpha| = \alpha_1 + \dots + \alpha_n, and the leading term, which is the a_{\alpha} x^{\alpha} with maximal \alpha under a . orders, such as the (lex order) where are compared by prioritizing the first variable (e.g., x_1 > x_2 > \dots > x_n), provide a total ordering on that is compatible with multiplication: if m_1 \geq m_2, then m \cdot m_1 \geq m \cdot m_2 for any m. This order ensures every nonzero polynomial has a unique leading term, facilitating algebraic manipulations. Examples include the univariate case over a k, such as k, the ring of polynomials in one indeterminate, and the bivariate case k[x, y], where elements are expressions like f(x, y) = 3x^2 y + 2x - 1. If the base R is an , then R[x_1, \dots, x_n] is also an ; in particular, polynomial rings over fields are .

Ideals in Rings

In a commutative ring R with identity, an ideal I is an additive subgroup of R such that for every x \in I and r \in R, the product rx \in I. This absorption property ensures that ideals are precisely the kernels of ring homomorphisms from R, making them central to quotient constructions in algebra. An ideal I \subseteq R is finitely generated if there exists a finite set \{g_1, \dots, g_k\} \subseteq I such that every element of I can be expressed as \sum_{i=1}^k r_i g_i for some r_i \in R. Such ideals arise naturally in many rings, but not all ideals need be finitely generated; for instance, in the polynomial ring \mathbb{Z}, the ideal (2, x) consists of all polynomials with even constant term and is generated by two elements, yet it cannot be generated by a single element, as any generator would need to divide both 2 and x in a way incompatible with the ring's structure. In contrast, rings like the polynomial ring in countably infinitely many variables over a field admit ideals that require infinitely many generators, such as the ideal generated by all the variables themselves, since any finite subset involves only finitely many variables and cannot produce a monomial in a new variable. A key concept related to ideal generation is that of ascending chains of ideals: a I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots in R. Stabilization occurs if there exists some n such that I_n = I_{n+1} = I_{n+2} = \cdots, meaning the chain eventually becomes constant; this property, when holding for every ascending chain, characterizes Noetherian rings and underpins finiteness results like Hilbert's basis theorem in the context of rings.

Statement and Formulation

Precise Statement

Hilbert's basis theorem asserts that if k is a , then every in the k[x_1, \dots, x_n], consisting of polynomials in n indeterminates over k, is finitely generated. This means that for any I \subseteq k[x_1, \dots, x_n], there exist finitely many polynomials f_1, \dots, f_m \in I such that every element of I can be expressed as a k- of the f_i with coefficients in k[x_1, \dots, x_n]. The theorem was originally formulated by in 1890, in the context of , where he proved that the ring of invariants of a over a under a linear group's action is finitely generated as an algebra. In his paper, Hilbert established this finiteness property more broadly for ideals in , laying the foundation for modern . A direct consequence of the theorem is that k[x_1, \dots, x_n] is a , meaning every in it is finitely generated.

Equivalent Versions

Hilbert's basis theorem is equivalent to the statement that the polynomial ring k[x_1, \dots, x_n] over a field k is a Noetherian ring, satisfying the ascending chain condition on ideals, as a ring is Noetherian if and only if every ideal is finitely generated. A related formulation concerns modules: every submodule of a free module of finite rank over k[x_1, \dots, x_n] is finitely generated. This follows from the more general principle that, over a Noetherian ring, every finitely generated module (including finite free modules) is itself Noetherian, meaning all its submodules are finitely generated. The theorem admits a broader extension to arbitrary commutative s: if R is a commutative Noetherian ring, then the R is also Noetherian, and by induction, so is R[x_1, \dots, x_n] for any finite n. This version generalizes the original, since fields are Noetherian rings. As a consequence, for any ideal I \subseteq k[x_1, \dots, x_n], the k[x_1, \dots, x_n]/I is finitely presented as a k-algebra, being isomorphic to a quotient of the polynomial ring by a finitely generated ideal.

Proofs

Proof Using Noetherian Rings

The proof of Hilbert's basis theorem proceeds by induction on the number of variables, establishing that the polynomial ring k[x_1, \dots, x_n] over a field k is Noetherian for all n \geq 0. For the base case n=1, consider the polynomial ring k. Let I be a nonzero ideal in k. Among elements of I, select a polynomial f of minimal degree d > 0. By scaling its leading coefficient (possible since k is a field), assume f is monic. For any g \in I, apply the division algorithm: g = q f + r where either r = 0 or \deg r < d. Then r = g - q f \in I. If r \neq 0, then \deg r < d, contradicting the minimality of \deg f. Thus, r = 0, so g \in (f) and I = (f). The zero ideal is finitely generated, so every ideal in k is finitely generated, proving k is Noetherian. Assume now that k[x_2, \dots, x_n] is Noetherian for some n \geq 2. To prove k[x_1, \dots, x_n] is Noetherian, first show that every monomial ideal (generated by monomials) is finitely generated; this is Dickson's lemma. Let J \subseteq k[x_1, \dots, x_n] be a monomial ideal, and fix x_1 as the variable for decomposition. Define J_0 = J \cap k[x_2, \dots, x_n], a monomial ideal in the lower-dimensional ring, hence finitely generated by the induction hypothesis. For i \geq 1, let J_i be the monomial ideal in k[x_2, \dots, x_n] generated by monomials m such that x_1^i m \in J. Then J = \langle J_0 \cup x_1 J_1 \cup x_1^2 J_2 \cup \cdots \rangle, and the J_i form an ascending chain J_1 \subseteq J_2 \subseteq \cdots. Since k[x_2, \dots, x_n] is Noetherian, this chain stabilizes: there exists s \geq 0 such that J_i = J_s for all i \geq s. Each J_i (for i = 0, \dots, s) is finitely generated, say by finite sets G_i, so J = \langle G_0 \cup x_1 G_1 \cup \cdots \cup x_1^s G_s \rangle is finitely generated. For a general ideal I \subseteq k[x_1, \dots, x_n], fix a monomial order > (e.g., lexicographic). The leading monomial \mathrm{LM}(f) of f \in I \setminus \{0\} is the highest term with respect to >. Let L = \{\mathrm{LM}(f) \mid f \in I \setminus \{0\}\}; then \langle L \rangle is a monomial ideal containing all leading monomials of I, hence finitely generated by Dickson's lemma, say by monomials m_1, \dots, m_r. For each i, choose f_i \in I such that \mathrm{LM}(f_i) = m_i. The key lemma states that if the leading monomials of generators form a monomial ideal containing all leading monomials of I, then I = (f_1, \dots, f_r). To verify, take g \in I. By the division algorithm with respect to the monomial order, divide g by f_1, \dots, f_r to obtain g = \sum q_i f_i + h, where h has no term divisible by any m_j (i.e., \mathrm{LT}(h) \notin \langle m_1, \dots, m_r \rangle). But \mathrm{LT}(g) \in \langle L \rangle = \langle m_1, \dots, m_r \rangle, so if h \neq 0, then \mathrm{LT}(h) \in \langle L \rangle, a contradiction. Thus, h = 0 and g \in (f_1, \dots, f_r). Hence, I is finitely generated, so k[x_1, \dots, x_n] is Noetherian. By induction, the theorem holds for all n.

Direct Ascending Chain Proof

A direct proof of Hilbert's basis theorem for the R in one indeterminate over a Noetherian ring R uses an ascending chain in R. Let I be an in R. For each non-negative integer n, define J_n = \{ r \in R \mid \exists f \in I \text{ with leading term } r x^n \}, more precisely, J_n is the set of coefficients r \in R such that there exists a polynomial f = r x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in I. Each J_n is an in R, and J_0 \subseteq J_1 \subseteq J_2 \subseteq \cdots, forming an ascending chain. Since R is Noetherian, this chain stabilizes: there exists m \geq 0 such that J_n = J_m for all n \geq m. Since R is Noetherian, each J_i for i = 0, \dots, m is finitely generated. For each i, select a F_i \subseteq I consisting of polynomials of exactly i whose leading coefficients generate J_i as an in R. Let F = \bigcup_{i=0}^m F_i, a . To show F generates I, proceed by on the of elements in I. The elements of less than or equal to m are generated by F since their leading coefficients are in the J_i's generated by those of F. For a g \in I of d > m, its leading a \in J_d = J_m, so a is an R- of leading coefficients of elements in F_m, say a = \sum b_j c_j where c_j are leading coefficients of f_j \in F_m. Then, consider \sum b_j f_j x^{d-m} - g, which has less than d and lies in I, so by , it is in (F), hence g \in (F). Thus, I is finitely generated by F, so R is Noetherian.

Generalizations and Extensions

To Principal Ideal Domains

Hilbert's basis theorem generalizes beyond fields to . If R is a , then the R[x_1, \dots, x_n] in any finite number of indeterminates is Noetherian: every in R[x_1, \dots, x_n] is finitely generated. This follows because every PID is itself Noetherian, as each ideal is principal and thus finitely generated by a single element. The original formulation for fields is a special case, since every field is a . However, R[x_1, \dots, x_n] is generally not a PID when R is a PID but not a field. For example, in \mathbb{Z}[x, y], the ideal (2, x, y) is finitely generated but not principal, as no single element divides both 2 and the indeterminates x, y. Even for one variable, \mathbb{Z} is Noetherian but not a PID; the ideal (2, x) cannot be principal, since any generator would need to divide both 2 (a constant) and x (of degree 1), which is impossible in \mathbb{Z}.

Hilbert's Nullstellensatz Connection

Hilbert's Nullstellensatz, published in 1893, builds directly on the foundation laid by his basis theorem from 1890, extending the algebraic insights into a profound link between ideals and geometric varieties. In his second major paper on , Hilbert utilized the finite generation of ideals guaranteed by the basis theorem to establish that vanishing sets of polynomials correspond to radical ideals, thereby bridging and . This connection was pivotal, as the basis theorem's assertion that rings over fields are Noetherian ensured that every ideal could be finitely generated, allowing for a precise algebraic description of geometric objects. The basis theorem plays a crucial role in the proof of the weak Nullstellensatz, which states that for an k, the maximal ideals of the k[x_1, \dots, x_n] are precisely of the form (x_1 - a_1, \dots, x_n - a_n) for some point a = (a_1, \dots, a_n) \in k^n. By the basis theorem, every , including any maximal ideal m, is finitely generated; the k[x_1, \dots, x_n]/m is then a of k that is finitely generated as a k-algebra. Since k is algebraically closed, this extension must coincide with k itself, implying that the images of the x_i under the quotient map are elements of k, and thus m is the of an at a point in k^n. This finite generation step is indispensable, as it reduces the analysis of arbitrary ideals to manageable finite sets of generators, facilitating the geometric interpretation. A key implication of this connection is the bijection between radical ideals in k[x_1, \dots, x_n] and algebraic sets (varieties) in affine space over k. Finitely generated ideals, by the basis theorem, define varieties as their common zero sets V(I), and the Nullstellensatz ensures that the ideal of polynomials vanishing on a variety V is the radical of the ideal generated by any finite set defining V. Specifically, for a radical ideal I, I(V(I)) = I, establishing a one-to-one correspondence that underpins much of classical algebraic geometry. This duality highlights how the basis theorem's algebraic finiteness enables the geometric rigidity of varieties over algebraically closed fields.

Applications

In Commutative Algebra

Hilbert's basis theorem implies that if R is a , then the R[x_1, \dots, x_n] is also Noetherian for any n \geq 0, establishing rings over fields or other Noetherian base rings as fundamental examples of Noetherian rings in . This Noetherian property ensures that ascending chains of ideals stabilize, facilitating the study of ideals and over these rings. In the context of graded over rings, the Noetherian condition has significant implications for the Hilbert function and Hilbert series. The Hilbert function h_M(d) of a finitely generated graded M over k[x_1, \dots, x_n], where k is a , counts the dimension of the degree-d component M_d. Due to the Noetherian structure, h_M(d) agrees with a polynomial of degree at most n-1 for sufficiently large d, known as the Hilbert polynomial, while the Hilbert series \sum_{d \geq 0} h_M(d) t^d is a rational function encoding the growth of the module. A key application arises in the of ideal decompositions: since rings k[x_1, \dots, x_n] are Noetherian, every admits a into a finite of primary , allowing the structure of to be analyzed through their primary components and associated primes. This finite decomposition is for understanding the and nilradical of in multivariate settings. rings over fields are Cohen-Macaulay rings, meaning their depth equals their dimension, and the Noetherian property from supports computations in local , where local modules with respect to relevant exhibit vanishing in degrees below the depth. This property simplifies over such rings, aiding in the classification of modules and the study of singularities in algebraic structures. The theorem's assurance of finite generation for ideals enables effective algorithms for ideal membership testing via Gröbner bases, which provide a canonical finite generating set allowing reduction of polynomials to determine containment in the ideal. This computational aspect underpins tools in computer algebra systems for solving systems of polynomial equations.

In Algebraic Geometry

Hilbert's basis theorem plays a foundational role in algebraic geometry by ensuring that polynomial ideals are finitely generated, which has direct implications for the structure of affine varieties. In the affine setting over an algebraically closed field k, an affine variety V \subseteq \mathbb{A}^n_k is defined as the zero set V(I) = \{ p \in \mathbb{A}^n_k \mid f(p) = 0 \ \forall f \in I \} of an ideal I \subseteq k[x_1, \dots, x_n]. The theorem guarantees that every such ideal I is finitely generated, say by polynomials f_1, \dots, f_m, implying that V(I) is the common zero set of only finitely many polynomials. This finite description is crucial, as it allows varieties to be specified parametrically with a finite number of equations, facilitating the study of their geometric properties and enabling the development of schemes as finite presentations over polynomial rings. For projective varieties, the theorem extends to graded rings, particularly through the finite of homogeneous ideals. Consider a X \subseteq \mathbb{P}^n_k with homogeneous ideal I(X) \subseteq k[x_0, \dots, x_n], which is finitely generated by homogeneous due to the Noetherian property of rings. The homogeneous coordinate ring S(X) = k[x_0, \dots, x_n]/I(X) is then a finitely generated graded k-algebra, and its Hilbert function h_S(d) = \dim_k S_d (the of the -d component) stabilizes for large d to yield the Hilbert P_S(t), a whose equals the of X and leading coefficient relates to the of X. This provides key invariants, such as the arithmetic genus for curves, and arises precisely because the finite allows of in each graded piece via a finite . The theorem also underpins dimension theory in through the Noetherian chain condition. The of an V(I) \subseteq \mathbb{A}^n_k is defined as the of the coordinate ring k[V(I)] = k[x_1, \dots, x_n]/I, which is the supremum of lengths of chains of in this ring. Since k[x_1, \dots, x_n] is Noetherian by iterated application of the basis theorem, all ascending chains of ideals stabilize, ensuring the dimension is well-defined and finite. For a P \subseteq k[x_1, \dots, x_n] of height h (the length of the longest chain of primes contained in P), Krull's height theorem implies that the dimension of k[x_1, \dots, x_n]/P is n - h, linking the geometric dimension of the corresponding irreducible variety to the of its defining . This relation extends to general varieties via their irreducible components. A concrete illustration is the coordinate ring of an affine curve. For an irreducible affine curve C \subseteq \mathbb{A}^n_k defined by a prime ideal I(C) of height n-1, the coordinate ring k[C] = k[x_1, \dots, x_n]/I(C) has Krull dimension 1 by the above dimension formula. Hilbert's basis theorem ensures I(C) is generated by finitely many polynomials, making k[C] a finitely generated k-algebra of dimension 1, which is integrally closed if C is nonsingular. For example, the affine elliptic curve C = V(y^2 - x^3 - x) \subseteq \mathbb{A}^2_k has coordinate ring k[x,y]/(y^2 - x^3 - x), finitely generated by the images of x and y.