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.
The ideal generated by a subset and principal ideals
Definition
The ideal generated by a subset and principal ideals.
For , define to be the intersection of all two-sided ideals of containing . This family is nonempty because it contains , and Ideal criteria and intersections of ideals shows that is an ideal. For , the ideal is written and is called principal.
Depends on
Used by
- For a field F, the ideal (x,y) in F[x,y] is not principal Counterexample
- Principal ideal domain Definition
- The monic greatest common divisor of two polynomials over a field Definition
- Every Euclidean domain is a principal ideal domain 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
- In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 14 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
- Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)