Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Example

R×{0}R\times\{0\} is the kernel of R×S→SR\times S\to S, so (R×S)/(R×{0})≅S(R\times S)/(R\times\{0\})\cong S.

Facts & Assumptions

Given: Rings R,SR,S and the coordinate projection p:R×S→Sp:R\times S\to S, p(r,s)=sp(r,s)=s.

[L1]

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

[L2]

A ring homomorphism preserves operations and identity (Ring homomorphism: additive, multiplicative, and required to send 11 to 11).

[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⁡fR/\ker f\cong\operatorname{im}f).

[L5]

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

Verification

technique · direct
1.1

Coordinatewise operations make pp a surjective ring homomorphism.

L1L2L3L4L5givenalgebra
2.1

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

step 1.1L1L2L3L4L5givenalgebra
3.1

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

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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