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

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The $p$ -adic completion map of the integers

Example

Let $p$ be a prime integer. The $(p)$ -adic completion map of $Z$ is $κ_{Z} ⁣ : Z \to {\hat{Z}}^{(p)}, n \mapsto (n m o d p^{r})_{r \geq 1} .$ It is injective, and its image consists of the ordinary integers viewed as compatible residue systems.

Facts & Assumptions

Given: A prime integer $p$ .

[L1]

The completion map for the $(p)$ -adic filtration sends an element to its compatible residue classes modulo $p^{r}$ (The $I$ -adic completion of a module).

[L2]

The kernel of the completion map is the intersection of the powers of the defining ideal (Kernel and universal property of adic completion).

Verification

technique · direct

1.1

Applying [L1] to $M = Z$ and $I = (p)$ gives the displayed formula for $κ_{Z}$ . Compatibility is automatic because reduction modulo $p^{r}$ followed by reduction modulo $p^{s}$ agrees with direct reduction modulo $p^{s}$ whenever $r \geq s$ .

L1algebra

1.2

By [L2], $\ker (κ_{Z}) = ⋂_{r \geq 1} p^{r} Z .$ If $n \neq 0$ , then for sufficiently large $r$ one has $p^{r} > ∣ n ∣$ , so $p^{r}$ cannot divide $n$ . Thus the intersection is $0$ , and $κ_{Z}$ is injective.

L2algebra

2.1

The image of $κ_{Z}$ is therefore exactly the copy of ordinary integers inside the completion, written componentwise as their residue systems modulo $p^{r}$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

J. S. Milne, A Primer of Commutative Algebra, Aside 24.7 (standard reference, not scraped)

The pp-adic completion map of the integers