PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-03

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

Statement

For a submodule $N \leq M$ , the canonical map

$π : M ⟶ M / N, π (m) : = m + N,$

is a surjective $R$ -module homomorphism and has kernel $N$ . Hence every submodule is the kernel of a module homomorphism.

Facts & Assumptions

Given: A left $R$ -module $M$ and a submodule $N \leq M$ .

[L1]

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

[L2]

The canonical projection of the underlying additive groups is a surjective group homomorphism (The canonical projection $π : G \to G / N$ , $π (g) = g N$ , 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 $m + N = N$ if and only if $m \in N$ (Submodule of a module, $x \in a H$ iff $a^{- 1} x \in H$ , and $a H = b H$ iff $a^{- 1} b \in H$ ).

Proof

technique · direct

1.1

The underlying additive map $π$ is a homomorphism and is surjective.

L2given

1.2

The quotient action gives $π (r m) = r m + N = r (m + N) = r π (m)$ .

L1given

1.3

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

L3L4given

2.1

Steps 1.1--1.2 show that $π$ 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

A module homomorphism vanishing on N factors uniquely through M/N Theorem

Dependency tree · two levels

16 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

McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)

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