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.
Left, right and two-sided ideals
Definition
Left, right and two-sided ideals.
Let be a ring. An additive subgroup is a left ideal when for every and , and a right ideal when for every such . A two-sided ideal, written , is both a left and a right ideal. In a commutative ring these three notions agree.
Depends on
Used by
- Annihilators, torsion elements and the torsion subset of a module Definition
- Prime ideals and maximal ideals in a commutative ring Definition
- The ideal generated by a subset and principal ideals Definition
- The quotient ring R/I with (r+I)(s+I)=rs+I Definition
- The sum I+J and product IJ of two-sided ideals Definition
- 2ℤ is an ideal of ℤ but is not a subring under the library's unital convention Example
- Left ideals are exactly the submodules of the regular left module _RR Example
- R×{0} is the kernel of R× S→ S, so (R× S)/(R×{0})≅ S Example
- Ideal criteria and intersections of ideals Lemma
- If I⊆ J are ideals of R, then J/I is an ideal of R/I Lemma
- 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
- A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring Theorem
- Correspondence theorem: ideals of R/I correspond to ideals of R containing I Theorem
- Every Euclidean domain is a principal ideal domain Theorem
- In a nonzero commutative ring, every proper ideal is contained in a maximal ideal Theorem
- Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal Theorem
- The kernel of a ring homomorphism is a two-sided ideal Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 10 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)