ExampleConstruction: AI-adaptedVerification: 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.

$R \times {0}$ is the kernel of $R \times S \to S$ , so $(R \times S) / (R \times {0}) ≅ S$

Example

$R \times {0}$ is the kernel of $R \times S \to S$ , so $(R \times S) / (R \times {0}) ≅ S$ .

Facts & Assumptions

Given: Rings $R, S$ and the coordinate projection $p : R \times S \to S$ , $p (r, s) = s$ .

[L1]

The product ring has coordinatewise operations (The product ring $R \times S$ with componentwise operations, its identity $(1_{R}, 1_{S})$ and its units $R^{\times} \times S^{\times}$ ).

[L2]

A ring homomorphism preserves operations and identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

[L3]

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

[L4]

The first ring isomorphism theorem gives quotient-by-kernel isomorphisms (First isomorphism theorem for rings: $R / \ker f ≅ im f$ ).

[L5]

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

Verification

technique · direct

1.1

Coordinatewise operations make $p$ a surjective ring homomorphism.

L1L2L3L4L5givenalgebra

2.1

Its kernel is exactly ${(r, 0) : r \in R} = R \times {0}$ , which is therefore an ideal.

step 1.1L1L2L3L4L5givenalgebra

3.1

The kernel calculation yields $(R \times S) / (R \times {0}) ≅ S$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Judson, Abstract Algebra: Theory and Applications, Ring Homomorphisms and Ideals (standard reference, not scraped)

R×{0}R×{0} is the kernel of R×S→SR×S→S, so (R×S)/(R×{0})≅S(R×S)/(R×{0})≅S