TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Universal property of the free module on a set

Statement

Let $R$ be a unital ring, $X$ a set, and $M$ a left $R$ -module. Every set map $u : X \to M$ extends uniquely to an $R$ -module homomorphism $\overset{ˉ}{u} : R^{(X)} \to M$ satisfying $\overset{ˉ}{u} (e_{x}) = u (x)$ . Explicitly, $\overset{ˉ}{u} (\sum_{x \in F} r_{x} e_{x}) = \sum_{x \in F} r_{x} u (x) .$

Facts & Assumptions

Given: A set map $u : X \to M$ .

[F1]

$R^{(X)}$ is the direct sum of copies of the regular module $R$ , with standard vectors $e_{x}$ and unique finite coordinate expressions (The free module on a set and its standard basis).

[L1]

A family of homomorphisms from the summands determines a unique homomorphism from their direct sum (Universal property of a direct sum of modules).

Proof

technique · constructive

1.1

For each $x \in X$ , define the homomorphism $f_{x} : R \to M$ by $f_{x} (r) = r u (x)$ .

givenconstruct

2.1

By [L1], the family $(f_{x})$ determines a unique homomorphism $\overset{ˉ}{u} : R^{(X)} \to M$ with $\overset{ˉ}{u} \circ ȷ_{x} = f_{x}$ .

step 1.1F1L1

3.1

Since $e_{x} = ȷ_{x} (1_{R})$ , one has $\overset{ˉ}{u} (e_{x}) = f_{x} (1_{R}) = u (x)$ , and additivity gives the displayed finite-sum formula.

step 2.1F1

4.1

Any homomorphism agreeing with $u$ on every $e_{x}$ agrees with $\overset{ˉ}{u}$ on every finite linear combination, hence on all of $R^{(X)}$ .

step 3.1F1

5.1

When $X = \emptyset$ , [F1] gives $R^{(X)} = 0$ and the unique map $0 \to M$ , so no nonempty choice is hidden. The construction and uniqueness prove the universal property.

step 1.1step 2.1step 3.1step 4.1F1discharge-construct∎

Depends on

Used by

Every module is a quotient of a free module Corollary
The standard basis and a universal map from R³ Example
Free modules are projective, with the exact choice boundary Theorem
Module categories have enough injectives Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 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)