DefinitionDefinition: 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 quotient ring $R / I$ with $(r + I) (s + I) = r s + I$

Definition

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

Let $I ⊴ R$ . Since $I$ is an additive subgroup of the abelian group $(R, +)$ , its additive cosets form the quotient group $R / I$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , Every subgroup of an abelian group is normal). Define

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

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

Depends on

Dependency tree · 2 levels

Sources

Ernst, An Inquiry-Based Approach to Abstract Algebra, Ideals and Quotient Rings