CorollaryStatement: Literature-sourcedProof: 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.
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.

Every maximal ideal of a commutative ring is prime

Statement

Every maximal ideal of a commutative ring is prime.

Facts & Assumptions

Given: A maximal ideal $M$ of a commutative ring $R$ .

[L1]

$R / M$ is a field when $M$ is maximal ( $R / M$ is a field if and only if $M$ is a maximal ideal).

[L2]

$R / P$ is a domain exactly when $P$ is prime ( $R / P$ is an integral domain if and only if $P$ is a prime ideal).

[L3]

Every field is an integral domain (Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring).

Proof

technique · direct

1.1

By [L1], $R / M$ is a field.

L1L2L3given

2.1

By [L3], that quotient is an integral domain.

step 1.1L1L2L3given

3.1

The domain conclusion of step 2.1 yields that $M$ is prime.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

17 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)