PropositionStatement: 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.

The canonical projection $R \to R / I$ is a surjective ring homomorphism with kernel $I$

The canonical projection $R \to R / I$ is a surjective ring homomorphism with kernel $I$ .

For $I ⊴ R$ , $π (r) = r + I$ has these properties.

Given: A ring $R$ and a two-sided ideal $I ⊴ R$ .

[L1]

[L2]

[L3]

[L4]

A kernel is the inverse image of the identity element (The kernel and image of a group homomorphism).

[L5]

The coset-equality criterion gives $r + I = 0 + I$ exactly when $- r \in I$ , hence exactly when $r \in I$ because $I$ is an additive subgroup ( $x \in a H$ iff $a^{- 1} x \in H$ , and $a H = b H$ iff $a^{- 1} b \in H$ ).

technique · direct

1.1L1L2givenalgebra

The identities $π (r + s) = π (r) + π (s)$ , $π (r s) = π (r) π (s)$ , and $π (1) = 1 + I$ show that $π$ is a ring homomorphism.

2.1step 1.1L3L4L5givenalgebra

By [L3] it is surjective; by [L4] and [L5], its kernel is exactly $I$ .

3.1step 2.1∎

Thus $\ker π = I$ .

Depends on