ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A semilocal Dedekind domain is a PID

Example

Assume the Axiom of Choice.

Let $R$ be a Dedekind domain with only finitely many maximal ideals. Then $R$ is a principal ideal domain.

Facts & Assumptions

Given: The Axiom of Choice and a Dedekind domain $R$ whose distinct maximal ideals are $m_{1}, \dots, m_{r}$ .

[L1]

Every nonzero ideal of $R$ has a unique prime-power factorization (Unique factorization of nonzero fractional ideals into prime powers).

[L2]

The Chinese remainder theorem solves simultaneous congruences modulo the pairwise comaximal powers $m_{i}^{n_{i} + 1}$ (Chinese remainder theorem for pairwise comaximal ideals).

[L3]

A Dedekind domain is a PID exactly when its class group is trivial (A Dedekind domain is a PID exactly when its class group is trivial).

Verification

technique · direct

1.1

Let $I = \prod_{i = 1}^{r} m_{i}^{n_{i}}$ be a nonzero ideal by [L1]. For each $i$ , choose $x_{i} \in m_{i}^{n_{i}} ∖ m_{i}^{n_{i} + 1}$ . By [L2], there exists $x \in R$ with $x \equiv x_{i} (m o d m_{i}^{n_{i} + 1})$ for each $i$ . Then $x$ has valuation exactly $n_{i}$ at $m_{i}$ for every $i$ , and there are no other primes to consider. Hence $(x) = I$ by uniqueness in [L1].

L1L2givenchoose

2.1

Every nonzero ideal of $R$ is therefore principal, so [L3] identifies $R$ as a PID.

L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. P. May, Notes on Dedekind Rings (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, §20 (standard reference, not scraped)