ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-02

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.
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For $n \geq 2$ , reduction $Z \to Z / n$ has kernel $n Z$ and realises $Z / n$ by the first isomorphism theorem

Example

For $n \geq 2$ , reduction $Z \to Z / n$ has kernel $n Z$ and realises $Z / n$ by the first isomorphism theorem.

Facts & Assumptions

Given: An integer $n \geq 2$ and $ρ_{n} : Z \to Z / n$ , $ρ_{n} (a) = [a]_{n}$ .

[L1]

The first isomorphism theorem identifies a group modulo a homomorphism kernel with its image (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[L2]

$(Z, +) / n Z$ is the congruence-class group $(Z / n, +)$ (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ ).

[L3]

A group homomorphism preserves the operation (Monoid homomorphism and group homomorphism).

[L4]

The kernel is the inverse image of the identity (The kernel and image of a group homomorphism).

[L5]

The integers form a commutative ring, hence an additive group (The integers form a commutative ring).

Verification

technique · direct

1.1

Since $[a + b]_{n} = [a]_{n} + [b]_{n}$ , $ρ_{n}$ is a homomorphism of additive groups.

L1L2L3L4L5givenalgebra

2.1

Its kernel is ${a : [a]_{n} = [0]_{n}} = n Z$ , and every residue class is $ρ_{n} (a)$ .

step 1.1L1L2L3L4L5givenalgebra

3.1

Therefore the kernel and image calculation yields $Z / n Z ≅ Z / n$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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

Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)

For n≥2n≥2, reduction Z→Z/nZ→Z/n has kernel nZnZ and realises Z/nZ/n by the first isomorphism theorem