Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

If I⊆J are ideals of R, then J/I is an ideal of R/I

Statement

If I⊆J are ideals of R, then J/I is an ideal of R/I.

Facts & Assumptions

Given: Two-sided ideals I⊆J⊴R.

[L1]

Ideals are additive subgroups with absorption (Left, right and two-sided ideals).

[L2]

R/I is a ring of additive cosets (For a two-sided ideal I, the additive cosets form a ring R/I with identity 1+I).

[L3]

The quotient map has kernel I (The canonical projection R→R/I is a surjective ring homomorphism with kernel I).

Proof

technique · direct
1.1

J/I={j+I:j∈J} is an additive subgroup of R/I because J is an additive subgroup.

L1L2L3givenalgebra
2.1

For r+I∈R/I and j+I∈J/I, both (r+I)(j+I)=rj+I and (j+I)(r+I)=jr+I lie in J/I.

step 1.1L1L2L3givenalgebra
3.1

Hence J/I is a two-sided ideal of R/I.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

12 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