CorollaryStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Every module is a quotient of a free module

Statement

For every left $R$ -module $M$ , the free module $R^{(M)}$ on its underlying set admits a canonical surjection $ε_{M} : R^{(M)} \to M$ , determined by $ε_{M} (e_{m}) = m$ . Consequently $M ≅ R^{(M)} / \ker ε_{M}$ .

Facts & Assumptions

Given: A left $R$ -module $M$ .

[L1]

Every set map from a basis set to a module extends uniquely to a homomorphism from the free module (Universal property of the free module on a set).

[F1]

The quotient module $F / K$ consists of additive cosets and carries the induced module operations (Quotient module $M / N$ with scalar multiplication on additive cosets).

Proof

technique · constructive

1.1

Apply [L1] to the identity set map on the underlying set of $M$ ; this defines $ε_{M} : R^{(M)} \to M$ with $ε_{M} (e_{m}) = m$ .

L1construct

2.1

Every $m \in M$ equals $ε_{M} (e_{m})$ , so $ε_{M}$ is surjective, including when $M = 0$ .

step 1.1

2.2

Define $ϕ : R^{(M)} / \ker ε_{M} \to M$ by $ϕ (x + \ker ε_{M}) = ε_{M} (x)$ . Equality of cosets makes this well defined, and [F1] makes it a homomorphism.

step 1.1F1construct

3.1

The map $ϕ$ is surjective by step 2.1 and injective because $ϕ (x + \ker ε_{M}) = 0$ exactly when $x \in \ker ε_{M}$ . Hence it is an isomorphism.

step 2.1step 2.2

4.1

Thus $M$ is canonically a quotient of a free module.

step 3.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 9 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)