TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.
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$R / M$ is a field if and only if $M$ is a maximal ideal

Statement

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

Here $R$ is commutative and $M$ is an ideal of $R$ .

Facts & Assumptions

Given: A commutative ring $R$ and a proper ideal $M ⊴ R$ .

[L1]

A maximal ideal has no proper intermediate ideal (Prime ideals and maximal ideals in a commutative ring).

[L2]

$R / M$ is a quotient ring with its usual coset operations (For a two-sided ideal $I$ , the additive cosets form a ring $R / I$ with identity $1 + I$ ).

[L3]

A field is a commutative ring in which every nonzero element is invertible (Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring).

[L4]

The definition of field requires a multiplicative inverse for each nonzero element (Field).

[L5]

The ideal criterion verifies ideals by subtraction and absorption (Ideal criteria and intersections of ideals).

Proof

technique · direct

1.1L1L2L3L4L5givenalgebra

If $M$ is maximal and $a \notin M$ , the set $J = {m + r a : m \in M, r \in R}$ is an ideal by the subtraction-and-absorption criterion, properly contains $M$ , and hence is $R$ ; thus $m + r a = 1$ for some $m \in M, r \in R$ , giving $(a + M) (r + M) = 1 + M$ .

2.1step 1.1L1L2L3L4L5givenchoose

If $R / M$ is a field and $M ⊊ J ⊴ R$ , choose $a \in J ∖ M$ ; an inverse $r + M$ of $a + M$ gives $a r - 1 \in M \subseteq J$ , while $a r \in J$ , so $1 \in J$ and $J = R$ .

3.1step 2.1∎

Hence $R / M$ is a field exactly when $M$ is maximal.

Depends on

Dependency tree · 2 levels

Sources

Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals