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.
is a field if and only if is a maximal ideal
Statement
is a field if and only if is a maximal ideal.
Here is commutative and is an ideal of .
Facts & Assumptions
Given: A commutative ring and a proper ideal .
A maximal ideal has no proper intermediate ideal (Prime ideals and maximal ideals in a commutative ring).
is a quotient ring with its usual coset operations (For a two-sided ideal , the additive cosets form a ring with identity ).
A field is a commutative ring in which every nonzero element is invertible (Every field is a commutative ring with ; it is an integral domain, and it is a commutative division ring).
The definition of field requires a multiplicative inverse for each nonzero element (Field).
The ideal criterion verifies ideals by subtraction and absorption (Ideal criteria and intersections of ideals).
Proof
If is maximal and , the set is an ideal by the subtraction-and-absorption criterion, properly contains , and hence is ; thus for some , giving .
If is a field and , choose ; an inverse of gives , while , so and .
Hence is a field exactly when is maximal.
Depends on
Used by
- Every maximal ideal of a commutative ring is prime Corollary
- Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points Corollary
- Under an integral extension, a prime is maximal if and only if its contraction is maximal Corollary
- A local ring is a nonzero commutative ring with a unique maximal ideal Definition
- F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
- For every integer n>1, nℤ is a maximal ideal of ℤ if and only if n is prime Example
- The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences Example
- A classical affine algebraic set has a unique finite irredundant decomposition Lemma
- A maximal ideal of an affine algebra has finite residue field over the base field Lemma
- Evaluation at a point has kernel (x₁-a₁,..., xₙ-aₙ) Lemma
- Points of an affine algebraic set correspond to maximal ideals of its coordinate ring Lemma
- A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem
- Assuming Choice, every field has an algebraic extension containing roots of all nonconstant base polynomials Theorem
- Every commutative Artinian ring is Noetherian Theorem
- Every prime ideal of an Artinian ring is maximal Theorem
- For a nonconstant p in F[x], the ideal (p) is maximal and F[x]/(p) is a field exactly when p is irreducible Theorem
Dependency tree · two levels
20 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)