Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02
How 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 sum and product of two-sided ideals
Definition
The sum and product of two-sided ideals.
For two-sided ideals , set
The empty sum in the definition of is .
Depends on
Used by
- A proper nonprojective scheme from glued projective spaces Counterexample
- The central reduction of the enveloping algebra at a central character Definition
- (R/I)⊗_R(R/J)≅ R/(I+J) for ideals of a commutative ring Example
- Class group of Q(sqrt 10) Example
- The tangent space of the parabola at a general point and the local parameter Example
- False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
- All initial forms define the tangent cone Lemma
- An ideal maximal among the non-finitely-generated ideals is prime Lemma
- Cotangent spaces commute with localization at a rational point Lemma
- The Noetherian minimal-prime induction split Lemma
- The radical of a product of ideals Lemma
- Vanishing sets of arbitrary sums Lemma
- Vanishing sets of finite products Lemma
- Primitive ideals are prime in the noncommutative sense Proposition
- Classical affine zero loci form the Zariski closed sets Theorem
- The sum and product of two-sided ideals are two-sided ideals Theorem
Dependency tree · two levels
3 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
- Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)