Alphabeta Math
TheoremStatement: 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.

Third isomorphism theorem for rings: (R/I)/(J/I)≅R/J

Statement

Third isomorphism theorem for rings: (R/I)/(J/I)≅R/J.

Facts & Assumptions

Given: Two-sided ideals I⊆J⊴R.

[L1]

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

[L2]

A ring modulo a kernel is isomorphic to the image (First isomorphism theorem for rings: R/ker⁡f≅im⁡f).

[L3]

Quotient projections are surjective ring homomorphisms (The canonical projection R→R/I is a surjective ring homomorphism with kernel I).

Proof

technique · direct
1.1L1L2L3givenconstruct

Define ϕ:R/I→R/J by ϕ(r+I)=r+J; it is a well-defined surjective ring homomorphism because I⊆J.

2.1step 1.1L1L2L3givenalgebra

The equality ϕ(r+I)=0+J holds exactly when r∈J, so ker⁡ϕ=J/I.

3.1step 2.1∎

The kernel and image computation gives (R/I)/(J/I)≅R/J.

Depends on

Dependency tree · 2 levels

Sources