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 quotient ring with
Definition
The quotient ring with .
Let . Since is an additive subgroup of the abelian group , its additive cosets form the quotient group (The quotient group and coset product , Every subgroup of an abelian group is normal). Define
The well-definedness and ring laws are established by Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal ↗ and For a two-sided ideal , the additive cosets form a ring with identity ↗.
Depends on
Used by
- A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
- homogeneous coordinate ring Definition
- Symmetric algebra of a vector space Definition
- The affine scheme of dual numbers Definition
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i Definition
- The coordinate ring of a classical affine algebraic set Definition
- The coordinate ring of an affine algebraic set Definition
- Universal enveloping algebra Definition
- A truncated polynomial local ring is Henselian Example
- k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
- Lift a Bezout identity for coprime residue factors Lemma
- Commutative rings form a reflective full subcategory of rings Theorem
- Every quotient and every localisation of a Noetherian ring is Noetherian 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
- For a two-sided ideal I, the additive cosets form a ring R/I with identity 1+I Theorem
- Localisation commutes with quotient rings: S⁻¹R/S⁻¹I≅ S̄⁻¹(R/I) Theorem
- Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal Theorem
Dependency tree · two levels
16 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
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Ideals and Quotient Rings (standard reference, not scraped)