Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

The kernel of a ring homomorphism is a two-sided ideal

Statement

The kernel of a ring homomorphism is a two-sided ideal.

Facts & Assumptions

Given: A ring homomorphism f:R→Sf:R\to S.

[L1]

A ring homomorphism preserves addition, multiplication, 00, and 11 (Ring homomorphism: additive, multiplicative, and required to send 11 to 11).

[L2]

Ring homomorphisms preserve additive inverses (A ring homomorphism satisfies f(0)=0f(0) = 0, f(−a)=−f(a)f(-a) = -f(a) and f(ma)=mf(a)f(ma) = m f(a) for m∈Zm \in \mathbb{Z}, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms).

[L3]

The group kernel is the inverse image of 00 in the additive groups (The kernel and image of a group homomorphism).

[L4]

A two-sided ideal is an additive subgroup with two-sided absorption (Left, right and two-sided ideals).

Proof

technique · direct
1.1

The additive-group kernel of ff is an additive subgroup of RR.

L1L2L3L4given
2.1

If x∈ker⁡fx\in\ker f and r∈Rr\in R, then f(rx)=f(r)f(x)=0f(rx)=f(r)f(x)=0 and f(xr)=0f(xr)=0.

step 1.1L1L2L3L4givenalgebra
3.1

Thus the additive and absorption properties make ker⁡f\ker f a two-sided ideal.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 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