Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03
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.

Module homomorphism and isomorphism, kernel, image and cokernel

Definition

For left RR-modules M,NM,N, a function f:M→Nf:M\to N is an RR-module homomorphism if

f(m+m′)=f(m)+f(m′)andf(rm)=rf(m)f(m+m')=f(m)+f(m')\quad\text{and}\quad f(rm)=rf(m)

for all m,m′∈Mm,m'\in M and r∈Rr\in R. It is a module isomorphism if it is a bijective module homomorphism.

Its kernel and image are

ker⁡f:={m∈M:f(m)=0N},im⁡f:={f(m):m∈M}.\ker f:=\{m\in M:f(m)=0_N\},\qquad\operatorname{im}f:=\{f(m):m\in M\}.

Once Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel ↗ establishes that the image is a submodule, the cokernel of ff is the quotient module coker⁡f:=N/im⁡f\operatorname{coker}f:=N/\operatorname{im}f.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 16 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