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

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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 $M\mathrel{\trianglelefteq}R$ .

[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 $1 \ne 0$ ; 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

1.1

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

L1L2L3L4L5givenalgebra

2.1

If $R/M$ is a field and $M\subsetneq J\mathrel{\trianglelefteq}R$ , choose $a\in J\setminus M$ ; an inverse $r+M$ of $a+M$ gives $ar-1\in M\subseteq J$ , while $ar\in J$ , so $1\in J$ and $J=R$ .

step 1.1L1L2L3L4L5givenchoose

3.1

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

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 13 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)

R/MR/MR/M is a field if and only if MMM is a maximal ideal