Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 quotient ring R/IR/I with (r+I)(s+I)=rs+I(r+I)(s+I)=rs+I

Definition

The quotient ring R/IR/I with (r+I)(s+I)=rs+I(r+I)(s+I)=rs+I.

Let I⊴RI\mathrel{\trianglelefteq}R. Since II is an additive subgroup of the abelian group (R,+)(R,+), its additive cosets form the quotient group R/IR/I (The quotient group G/NG/N and coset product (gN)(hN)=ghN(gN)(hN)=ghN, Every subgroup of an abelian group is normal). Define

(r+I)(s+I):=rs+I.(r+I)(s+I):=rs+I.

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 II, the additive cosets form a ring R/IR/I with identity 1+I1+I ↗.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 14 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