ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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For every integer $n>1$ , $n\mathbb Z$ is a maximal ideal of $\mathbb Z$ if and only if $n$ is prime

Example

For every integer $n>1$ , $n\mathbb Z$ is a maximal ideal of $\mathbb Z$ if and only if $n$ is prime.

Facts & Assumptions

Given: An integer $n>1$ .

[L1]

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

[L2]

A quotient is a field exactly for a maximal ideal ( $R/M$ is a field if and only if $M$ is a maximal ideal).

[L3]

A maximal ideal in a commutative ring is prime (Every maximal ideal of a commutative ring is prime).

[L4]

An integer greater than $1$ is prime exactly when it has the Euclid property (For an integer $p > 1$ : $p$ is prime if and only if, for all integers $a$ and $b$ , $p \mid ab$ implies $p \mid a$ or $p \mid b$ ).

[L5]

$\mathbb Z/p$ is a field for prime $p$ (For every prime $p$ , the two operations on $\mathbb{Z}/p$ make it a field).

[L6]

$\mathbb Z$ is a commutative ring (The integers form a commutative ring).

Verification

technique · direct

1.1

If $n$ is prime, [L5] and [L1] make $\mathbb Z/n\mathbb Z$ a field, so [L2] makes $n\mathbb Z$ maximal.

L1L2L3L4L5L6given

2.1

If $n\mathbb Z$ is maximal, [L3] makes it prime; applying this to a factorisation $n=ab$ gives the Euclid property and [L4] makes $n$ prime.

step 1.1L1L2L3L4L5L6givenalgebra

3.1

The two implications prove the example.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 97 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals (standard reference, not scraped)

For every integer n>1n>1n>1, nZn\mathbb ZnZ is a maximal ideal of Z\mathbb ZZ if and only if nnn is prime