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.
Prime ideals and maximal ideals in a commutative ring
Definition
Prime ideals and maximal ideals in a commutative ring.
Let be a commutative ring. A proper ideal is prime when implies or . A proper ideal is maximal when there is no proper ideal strictly between and ; equivalently, it is a maximal element of the poset of proper ideals ordered by inclusion.
Depends on
Used by
- The zero ideal of ℤ is prime but not maximal Counterexample
- Null rational sequences form a maximal ideal in the ring of rational Cauchy sequences Example
- 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
- In a nonzero commutative ring, every proper ideal is contained in a maximal ideal Theorem
- R/M is a field if and only if M is a maximal ideal Theorem
- R/P is an integral domain if and only if P is a prime ideal Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 15 results over 11 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)