The Goldman-Krull theorem

D. J. Bernstein

19980225

Goldman domains. A domain G is a Goldman domain if G[β] = fof G for some β with β^{−1} ∈ G. For example, any field is a Goldman domain.

Theorem 1. Let F be a domain. Then F [x] is not a Goldman domain.

Proof. Suppose that F [x][u^{−1}] is a field with u ∈ F [x]. The polynomial 1−ux is nonzero, so it has an inverse in F [x][u^{−1}]. Write (1−ux)^{−1} in the form fu^{−n} with f ∈ F [x] and n ∈ N. Then 1 = 1−(ux)^{n}+(1−ux)fx^{n} ∈ (1 − ux)F [x]; so u = 0. Contradiction.

Theorem 2. Let F be a subfield of a field K. Let α be an element of K. If F [α] is a Goldman domain then α is F -integral.

Proof. If α is not F -integral then F [α] is isomorphic to the polynomial ring F [x]. Apply Theorem 1.

Jacobson rings. A prime ideal P of a ring R is a Goldman ideal if R/P is a Goldman domain. A ring is a Jacobson ring if all of its Goldman ideals are maximal ideals. Examples: any field is a Jacobson ring; Z is a Jacobson ring.

Theorem 3. Let R be a Jacobson ring. Let h be a ring homomorphism from R[x] onto a Goldman domain G. Then G is a field. Furthermore, h(R) is a field, and G is finitely generated as an h(R)-module.

Proof. Define α = h(x) and D = h(R). Define F = fof D and K = fof G. Find β ∈ K such that K = G[β] and β^{−1} ∈ G. Then G = D[α] and K = D[α][β]. Now F [α][β] = K, with β^{−1} ∈ F [α], so F [α] is a Goldman domain. By Theorem 2, α is F -integral. Thus F [α] is a field, finitely generated as an F -module. But β^{−1} ∈ F [α] so β ∈ F [α]. Thus β is F -integral. Clear denominators: α and β are D[γ]-integral for some γ with γ^{−1} ∈ D. Thus K = D[γ][α][β] is D[γ]-integral; so D[γ] is a field. Thus D is a Goldman domain. Since R is a Jacobson ring, D is a field, so D = F. Thus G = F [α] is a field, finitely generated as a D-module.

Theorem 4. Let R be a Jacobson ring. Let h be a ring homomorphism from R[x_{1}, x_{2}, …, x_{n}] onto a field G. Then h(R) is a field, and G is finitely generated as an h(R)-module.

Proof. There is nothing to prove for n = 0. Otherwise define A = R[x_{1}]. By Theorem 3, A is a Jacobson ring. By induction, h(A) is a field, and G is finitely generated as an h(A)-module. By Theorem 3, h(R) is a field, and h(A) is finitely generated as an h(R)-module; hence G is finitely generated as an h(R)-module.