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

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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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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 $1 \ne 0$ ; 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

For every integer n>1, nℤ is a maximal ideal of ℤ if and only if n is prime Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 14 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)