The Krull-Akizuki situation: theorems D. J. Bernstein 19951120 Krull dimension. Let R be a Noetherian ring. The Krull dimension of R, written dim R, is the supremum of all lengths of strict chains of prime ideals of R. (Recall that the length of a chain P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{t} is t, the number of inclusions.) For example, dim R = 0 means that every prime ideal is maximal. If R is a domain (i.e., if 0R is a prime ideal), then dim R ≤ 1 means that every nonzero prime ideal is maximal. Artin rings. A Noetherian ring R is Artin if dim R = 0. The Krull-Akizuki situation. Let K be a field. Let F be a subfield of K, let A be a subring of K, and let R be a subring of F contained in A. We say that R ⊆ A, F ⊆ K are in the Krull-Akizuki situation if (1) F is the field of fractions of R, (2) dim_{F} K is finite, (3) R is Noetherian, and (4) dim R ≤ 1. Theorem F1. Let R be a domain. Let k be a subfield of R. If dim_{k} R is finite then R is a field. Theorem L1. Let R be a ring. Let P be a maximal ideal of R. Let I be a finitely generated ideal of R. Then len_{R}(I/P I) is finite. Theorem L2. Let R be a ring, P a finitely generated maximal ideal of R. Then len_{R}(R/P^{n}) is finite. Theorem A1. Let R be an Artin ring. If P ≠ P′ are prime ideals of R then P and P′ are coprime. Theorem A2. Let R be an Artin ring. Then the set of prime ideals of R is finite. Furthermore R is isomorphic to ∏_{P} (R/P^{e(P)}) for certain exponents e(P). Theorem A3. Let R be an Artin ring. Then len_{R} R is finite. Theorem C1. Let R be a Noetherian ring with dim R ≤ 1. Let I be a nonzero ideal of R. Then R/I is Artin. Theorem V1. Let R be a Noetherian domain with dim R ≤ 1. Let F be the field of fractions of R. Let V be a vector space over F. Let M be a finitely generated R-submodule of V such that dim_{F} MF is finite. Let a be a nonzero element of R. Then len_{R}(R/a) is finite, and len_{R}(M/aM) = (dim_{F} MF) len_{R}(R/a). Theorem V2. Let R be a Noetherian domain with dim R ≤ 1. Let F be the field of fractions of R. Let V be a finite-dimensional vector space over F. Let M be an R-submodule of V. Let a be a nonzero element of R. Then len_{R}(M/aM) is finite. Theorem V3. Let R be a Noetherian domain with dim R ≤ 1. Let F be the field of fractions of R. Let V be a finite-dimensional vector space over F. Let M be an R-submodule of V. Let I be a nonzero ideal of R. Then len_{R}(M/IM) is finite. Theorem K1. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Let I be a nonzero ideal of R. Then len_{R}(A/IA) is finite. Theorem K2. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Let J be a nonzero ideal of A. Then J ∩ R has a nonzero element. Theorem K3. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Then A is Noetherian. Theorem K4. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Let Q be a nonzero prime ideal of A. Then Q ∩ R is a nonzero prime ideal of R. Theorem K5. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Then dim A ≤ 1.