ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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$Z^{2} / ⟨ (2, 0), (0, 3) ⟩$ is cyclic of order six

Example

The quotient $Z^{2} / ⟨ (2, 0), (0, 3) ⟩$ is cyclic of order six, generated by the class of $(1, 1)$ .

Facts & Assumptions

Given: The displayed relation matrix and elementary integer row and column operations.

[L1]

$R / (a b) ≅ R / (a) \oplus R / (b)$ when $a$ and $b$ are coprime (Coprime cyclic quotients over a PID split by the Chinese remainder map).

Verification

technique · direct

1.1

The relation matrix is $diag (2, 3)$ . Since $2$ and $3$ are coprime, elementary integer row and column operations using $3 - 2 = 1$ transform it to $diag (1, 6)$ , which is its Smith form.

givenalgebra

2.1

Directly, the quotient is $Z / 2 \oplus Z / 3$ , and [L1] identifies this with $Z / 6$ ; the unit Smith factor in step 1.1 contributes no summand.

step 1.1L1

3.1

The class of $(1, 1)$ has order the least common multiple of $2$ and $3$ , namely $6$ , so it generates the order-six quotient. Its lower positive multiples are nonzero because neither coordinate has simultaneously reached its relation.

step 2.1algebra∎

Depends on

Coprime cyclic quotients over a PID split by the Chinese remainder map

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

M. Brussel, Finitely Generated Modules over a PID, Smith and CRT examples (standard reference, not scraped)

Z2/⟨(2,0),(0,3)⟩Z2/⟨(2,0),(0,3)⟩ is cyclic of order six