Alphabeta MathHow 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.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
For every integer , is a maximal ideal of if and only if is prime
Example
For every integer , is a maximal ideal of if and only if is prime.
Facts & Assumptions
Given: An integer .
is the quotient ring (For every , the congruence-class ring is the quotient ring ).
A quotient is a field exactly for a maximal ideal ( is a field if and only if is a maximal ideal).
A maximal ideal in a commutative ring is prime (Every maximal ideal of a commutative ring is prime).
An integer greater than is prime exactly when it has the Euclid property (For an integer : is prime if and only if, for all integers and , implies or ).
is a field for prime (For every prime , the two operations on make it a field).
is a commutative ring (The integers form a commutative ring).
Verification
If is prime, [L5] and [L1] make a field, so [L2] makes maximal.
If is maximal, [L3] makes it prime; applying this to a factorisation gives the Euclid property and [L4] makes prime.
The two implications prove the example.
Depends on
- For every $n\in\mathbb N$, the congruence-class ring $\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$
- $R/M$ is a field if and only if $M$ is a maximal ideal
- Every maximal ideal of a commutative ring is prime
- 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 $p \mid b$
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- The integers form a commutative ring
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
44 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, Maximal and Prime Ideals (standard reference, not scraped)