ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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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The $p$ -adic integers as an inverse limit and as a completion

Example

Let $p$ be a prime integer. The inverse system $\dots \to Z / p^{3} Z \to Z / p^{2} Z \to Z / p Z$ with the natural reduction maps has inverse limit $Z_{p} : = {\underset{\leftarrow}{l i m}}_{n} Z / p^{n} Z .$ This module is canonically the $(p)$ -adic completion of $Z$ .

Facts & Assumptions

Given: A prime integer $p$ .

[L2]

The $(p)$ -adic completion of $Z$ is ${\hat{Z}}^{(p)} = {\underset{\leftarrow}{l i m}}_{n} Z / p^{n} Z$ with completion map $m \mapsto (m m o d p^{n})_{n}$ (The $I$ -adic completion of a module).

[L3]

The completion map has kernel $⋂_{n \geq 0} p^{n} Z$ (Kernel and universal property of adic completion).

Verification

technique · direct

1.1L2

By [L2], applying the adic-completion definition to the ring $Z$ and the ideal $(p)$ gives exactly the inverse system displayed above. Therefore its inverse limit is canonically the $(p)$ -adic completion of $Z$ .

1.2L3algebra

Under this identification, the completion map is the familiar residue map $Z \to Z_{p}, m \mapsto (m m o d p^{n})_{n \geq 1} .$ Its kernel is $⋂_{n \geq 0} p^{n} Z$ , which is $0$ because the only integer divisible by every power of $p$ is $0$ .

2.1step 1.1step 1.2∎

So the compatible-residue construction of $Z_{p}$ and the $(p)$ -adic-completion construction agree canonically.

Depends on

Used by

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Dependency tree · two levels

6 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., §22.5 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, §24 (standard reference, not scraped)

The pp-adic integers as an inverse limit and as a completion