ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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$Z / 12 Z$ splits as the product of its two local Artinian factors

Example

The ring $Z / 12 Z$ decomposes as $Z / 12 Z ≅ Z / 3 Z \times Z / 4 Z .$ The two factors are local Artinian rings, and the two prime ideals of $Z / 12 Z$ are the pullbacks of the two coordinate prime ideals.

Facts & Assumptions

Given: The ring $Z / 12 Z$ .

Verification

technique · direct

1.1givenalgebra

The ideals $(3)$ and $(4)$ of $Z$ are comaximal, so Chinese remainder theorem for pairwise comaximal ideals gives $Z / 12 Z ≅ Z / 3 Z \times Z / 4 Z$ . The first factor is a field, hence local, and the second has unique proper nonzero ideal $(2)$ , so it is also local.

2.1step 1.1givencasesalgebra

Let $P$ be a prime ideal of $Z / 3 Z \times Z / 4 Z$ . Since $(1, 0) (0, 1) = (0, 0) \in P$ , primality forces $(1, 0) \in P$ or $(0, 1) \in P$ . In the first case every $(a, 0) = (a, 0) (1, 0)$ lies in $P$ , so $P = Z / 3 Z \times Q$ for an ideal $Q$ of $Z / 4 Z$ ; primality then forces $Q = (2)$ . In the second case $P = 0 \times Z / 4 Z$ . Under the inverse of the isomorphism in step 1.1, these two primes pull back to $(2) / (12)$ and $(3) / (12)$ in $Z / 12 Z$ .

3.1step 1.1step 2.1∎

So $Z / 12 Z$ is exhibited concretely as the product of its two local Artinian factors $Z / 3 Z$ and $Z / 4 Z$ , and its two prime ideals are exactly the two coordinate pullbacks found in step 2.1.

Depends on

Chinese remainder theorem for pairwise comaximal ideals

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise (1.14) (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, Theorem 16.7 (standard reference, not scraped)

Z/12ZZ/12Z splits as the product of its two local Artinian factors