Alphabeta MathHow 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.
A module homomorphism factors as quotient by its kernel followed by inclusion of its image
Example
For a module homomorphism , the quotient map , the first-isomorphism isomorphism , and the inclusion together form the canonical epimorphism-monomorphism factorization of .
Facts & Assumptions
Given: A module homomorphism .
Modules over a ring form an abelian category (Modules over a ring form an abelian category).
The first isomorphism theorem for modules identifies with (First isomorphism theorem for modules: ).
Verification
The quotient map is the coimage projection of , and the inclusion is its image inclusion.
The isomorphism from [L2] identifies those two middle objects, and its composite with and the inclusion is exactly . So the usual module factorization is the categorical coimage-image factorization.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Gautam Tamme, Algebra II Lecture 9 (standard reference, not scraped)