Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

The canonical map M→M/NM\to M/N is a surjective module homomorphism with kernel NN; thus every submodule is a kernel

Statement

For a submodule N≤MN\le M, the canonical map

π:M⟶M/N,π(m):=m+N,\pi:M\longrightarrow M/N,\qquad \pi(m):=m+N,

is a surjective RR-module homomorphism and has kernel NN. Hence every submodule is the kernel of a module homomorphism.

Facts & Assumptions

Given: A left RR-module MM and a submodule N≤MN\le M.

[L1]

The quotient action is a well-defined module action (The quotient action is well defined and makes M/NM/N a module).

[L2]

The canonical projection of the underlying additive groups is a surjective group homomorphism (The canonical projection π:G→G/N\pi:G\to G/N, π(g)=gN\pi(g)=gN, is a surjective group homomorphism).

[L3]

A module homomorphism preserves addition and scalar multiplication, and its kernel is the inverse image of zero (Module homomorphism and isomorphism, kernel, image and cokernel).

[L4]

The additive group of a submodule is closed under inverses, and the coset criterion consequently gives m+N=Nm+N=N if and only if m∈Nm\in N (Submodule of a module, x∈aHx\in aH iff a−1x∈Ha^{-1}x\in H, and aH=bHaH=bH iff a−1b∈Ha^{-1}b\in H).

Proof

technique · direct
1.1

The underlying additive map π\pi is a homomorphism and is surjective.

L2given
1.2

The quotient action gives π(rm)=rm+N=r(m+N)=rπ(m)\pi(rm)=rm+N=r(m+N)=r\pi(m).

L1given
1.3

Since π(m)=0M/N=N\pi(m)=0_{M/N}=N exactly when m∈Nm\in N, its kernel is NN.

L3L4given
2.1

Steps 1.1--1.2 show that π\pi is a surjective module homomorphism.

step 1.1step 1.2L3
3.1

Together with step 1.3, this proves the claim and shows that every submodule is a kernel.

step 1.3step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 13 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