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.
Ideal criteria and intersections of ideals
Statement
Ideal criteria and intersections of ideals.
A nonempty subset is a two-sided ideal exactly when it is closed under and under for all , . Any intersection of two-sided ideals is a two-sided ideal, with the empty intersection equal to .
Facts & Assumptions
Given: A ring and a subset .
A two-sided ideal is an additive subgroup closed under left and right multiplication by ring elements (Left, right and two-sided ideals).
A nonempty subset closed under is a subgroup (One-step subgroup test: a nonempty is a subgroup iff for all ; the identity and the inverses of are then those of ).
A ring has additive inverses and distributive multiplication (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).
Proof
Closure under is exactly the additive subgroup criterion, and the two absorption conditions then give the ideal criterion.
An intersection has subtraction closure and both absorption properties because each member ideal has them; for the empty family the intersection is .
The criterion and intersection assertion follow.
Depends on
- Left, right and two-sided ideals
- One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $gh^{-1} \in H$ for all $g, h \in H$; the identity and the inverses of $H$ are then those of $G$
- Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
Used by
- The zero ideal of ℤ is prime but not maximal Counterexample
- The ideal generated by a subset and principal ideals Definition
- 2ℤ is an ideal of ℤ but is not a subring under the library's unital convention Example
- If S is a subring and I is an ideal of R, then S+I is a subring, I is an ideal of S+I, and S∩ I is an ideal of S Lemma
- In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra 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
- The sum and product of two-sided ideals are two-sided ideals Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 16 results over 12 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)