ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-17

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.

The ring $(Z / 2)^{N}$ is not Noetherian

Example

The product ring $R = (Z / 2)^{N}$ is not Noetherian: its ideals generated by the first finitely many coordinate idempotents form a strict ascending chain. See Left and right Noetherian rings.

Facts & Assumptions

Given: The hypotheses and objects in the Example.

[L1]

A unital ring $R$ is left Noetherian when its left regular module $_{R} R$ is Noetherian, and right Noetherian when the right regular module $R_{R}$ is Noetherian. Unqualified “Noetherian ring” means left Noetherian here; the side is stated whenever both notions occur. (Left and right Noetherian rings).

[L2]

Let $R$ be a unital ring and $(M_{i})_{i \in I}$ a family of left $R$ -modules (def-left-and-right-modules). Their direct product is the module $\prod_{i \in I} M_{i}$ with coordinatewise operations. The support of $m = (m_{i})$ is ${i \in I : m_{i} \neq 0}$ , and the direct sum is the submodule $⨁_{i \in I} M_{i} = {m \in \prod_{i \in I} M_{i} : supp (m) is finite}$ (def-submodule). (The direct sum of an indexed family of modules).

[L3]

For every $n \in N$ , with addition and multiplication as in def-addition-and-multiplication-modulo-n: 1. $(Z / n, +, [0]_{n})$ is an abelian group (def-group), with $- [a]_{n} = [- a]_{n}$ ; 2. $(Z / n, \cdot, [1]_{n})$ is a commutative monoid (def-semigroup-and-monoid); 3. multiplication distributes over addition on both sides. (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

Verification

technique · direct

1.1L1L2L3givenalgebra

In $R = (Z / 2)^{N}$ , let $e_{i}$ have value $1$ in coordinate $i$ and $0$ elsewhere, and set $I_{n} = R e_{0} + \dots + R e_{n}$ . Then $I_{n}$ consists exactly of the sequences supported in ${0, \dots, n}$ .

2.1step 1.1givenalgebra

Since $e_{n + 1} \in I_{n + 1} ∖ I_{n}$ , the chain $I_{0} < I_{1} < I_{2} < \dots$ is strictly ascending; hence the regular module is not Noetherian.

3.1step 2.1givenalgebra∎

The union $⋃_{n} I_{n}$ is only the finite-support ideal; for example the multiplicative identity $(1, 1, \dots)$ belongs to the product ring but not to that union. Thus the witness is genuinely an ideal chain in the full product ring. This proves the stated claim.

Depends on

Used by

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Dependency tree · two levels

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The ring (Z/2)N(Z/2)N is not Noetherian