Unique factorization in Dedekind domains: theorems D. J. Bernstein 19950917 Let R be a subring of some field F. Language. The reader should be familiar with the following concepts: prime ideal, maximal ideal, maximal element of a collection, finitely generated ideal, invertible ideal, module sum, module scalar product, module product, module ratio. Facts about Dedekind domains. We say that R is Dedekind if (1) dim R ≤ 1, (2) R is integrally closed, and (3) R is Noetherian. These conditions mean the following for a nonzero ideal I of R.

(1) dim R ≤ 1: if I is prime then I is a maximal ideal.

(2) R is integrally closed: if I is finitely generated then I ÷ I = R.

(3) R is Noetherian: every nonempty collection of ideals has a maximal element; hence I is finitely generated. Outline. What follows is a streamlined version of Krull’s 1928 proof that Dedekind domains have unique factorization into prime ideals. Define N as the collection of nonzero ideals with nontrivial inverse. For a nonzero ideal I, I ∉ N means that R = R ÷ I, and I ∈ N means that zI ⊆ R for some z ∉ R. Theorem G1. Let I be a nonzero ideal. If I is invertible and I ≠ R then I ∈ N. Theorem G2. If P is a maximal element of N then P is prime. Theorem G3. Let I, J, P be ideals, with P prime. If IJ ⊆ P then I ⊆ P or J ⊆ P. Theorem M1. Assume dim R ≤ 1. Let P, P_{1}, P_{2}, …, P_{r} be nonzero prime ideals of R. If P_{1}P_{2} … P_{r} ⊆ P then P = P_{i} for some i. Theorem I1. Assume that R is integrally closed. Let I and J be nonzero ideals, with I finitely generated. If J ∈ N then I ⊂ I(R ÷ J). Theorem I2. Assume that R is integrally closed. Let I be a nonzero finitely generated ideal. If I is a maximal ideal and I ∈ N then I is invertible. Theorem D1. Assume that R is Dedekind. Let I be a nonzero ideal. If I is invertible and I ≠ R then I is contained in an invertible maximal ideal. Theorem D2. Assume that R is Dedekind. Let I be a nonzero ideal. If I is invertible then I is a product of invertible maximal ideals. Theorem D3. Assume that R is Dedekind. If P is a nonzero prime ideal then P is invertible. Theorem D4. Assume that R is Dedekind. If I is a nonzero ideal then I is invertible.