LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A field has only the zero ideal and itself, hence is Noetherian

Statement

Let $K$ be a field (Field). Then the only ideals of $K$ are the zero ideal $(0)$ and $K$ itself (Left, right and two-sided ideals, The ideal generated by a subset and principal ideals); consequently every ideal of $K$ is finitely generated, and $K$ is a Noetherian ring (Left and right Noetherian rings, Noetherian modules: every submodule is finitely generated). No choice principle is used.

Facts & Assumptions

Given: A field $K$ and an ideal $I \subseteq K$ .

[F1]

Field: in a field every nonzero element $a$ has a multiplicative inverse $a^{- 1}$ with $a a^{- 1} = 1$ , and $1 \neq 0$ .

[F2]

Left, right and two-sided ideals: an ideal $I \subseteq K$ is an additive subgroup closed under multiplication by elements of $K$ , so $r a \in I$ for all $r \in K$ , $a \in I$ ; hence $I = K$ as soon as $1 \in I$ .

[F3]

The ideal generated by a subset and principal ideals: for $a \in K$ the ideal $(a)$ is the intersection of all ideals containing $a$ ; in particular $(0) = {0}$ is generated by $0$ and $K = (1)$ is generated by $1$ , so both are generated by a single element.

[F4]

Left and right Noetherian rings and Noetherian modules: every submodule is finitely generated: a ring is Noetherian when its left regular module is Noetherian, meaning that every left submodule is finitely generated. For the commutative field $K$ , these submodules are precisely its ideals.

Proof

technique · direct

1.1F1F2given

A nonzero ideal is everything: if $I \neq (0)$ choose $a \in I$ with $a \neq 0$ ; by [F1] $a$ is invertible with inverse $a^{- 1} \in K$ , and since $I$ is closed under multiplication by elements of $K$ , $1 = a^{- 1} a \in I$ by [F2]; then $x = x \cdot 1 \in I$ for every $x \in K$ by [F2] again, so $I = K$ .

2.1step 1.1F3

The ideal list: by step 1.1 every ideal of $K$ is either $(0)$ or $K$ ; the zero ideal is generated by the single element $0$ and $K = (1)$ is generated by the single element $1$ , so every ideal of $K$ is finitely generated.

3.1step 2.1F4given∎

Conclusion: by step 2.1 every ideal of $K$ is finitely generated, so [F4] makes $K$ a Noetherian ring. The argument used only the field axioms, the ideal axioms and the two-element list of ideals, so it invokes no choice principle.

Depends on

Used by

Euler characteristic in a proper flat family is locally constant Corollary
Finite-dimensional coherent cohomology over a field Corollary
Global functions on geometrically connected and geometrically reduced proper schemes Corollary
The dimension of a complete linear system Corollary
A cusp family defeats the missing source hypothesis Counterexample
The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
The Lie algebra does not detect nonsmooth group schemes Counterexample
Base points and base-point-free linear systems Definition
Canonical bundle and canonical divisors Definition
Complete linear system Definition
Delta invariant of a curve singularity Definition
Divisors on a smooth proper curve Definition
Genus and arithmetic genus of a curve Definition
Ramification points, branch points and unramifiedness Definition
The intrinsic Zariski tangent space Definition
The Riemann-Roch dimension l(D) Definition
A nondegenerate projective quadric Example
All twists on the projective line Example
An upper jump of h0 in a flat projective family Example
Pulling a divisor back along the cusp normalization Example
Blowups of integral projective complex schemes are projective Lemma
Dense relative-dimension strata in flat finitely presented fibres Lemma
Finite-dimensionality of the Riemann-Roch space Lemma
Finite-type field extensions with zero Ω Lemma
Flat field extension commutes with coherent cohomology Lemma
Free differentials imply regularity in characteristic zero Lemma
Local fibre dimension equals local ring dimension plus residue transcendence degree Lemma
Local fibre-dimension bound from polynomial quasi-finiteness Lemma
Local support and index bound for the different of a curve map Lemma
Rational maps from a smooth curve to a proper scheme are morphisms Lemma
Smooth closed immersion is regular with exact conormal sequence Lemma
Smooth immersions, their conormal sequence, deformation charts and smooth sections Lemma
Smooth orbits are locally closed and their orbit maps are faithfully flat over every field Lemma
The exact sequence for adding one point to a divisor Lemma
The quotient by a maximal line subbundle is locally free Lemma
Unramified residue extensions are finite separable Lemma
Upper semicontinuity of proper fibre dimension Lemma
A base-point-free linear system defines a morphism to projective space Theorem
Degree of the coherent Hilbert polynomial Theorem
Euler characteristic is a Hilbert polynomial Theorem

…and 3 more results.

Dependency tree · two levels

11 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

Stacks Project, Algebra, Section 10.31 (tag 00FM) and Lemma 10.31.3 (tag 00FO) (standard reference, not scraped)