DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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The free module on a set and its standard basis

Definition

For a unital ring $R$ and a set $X$ , the free left $R$ -module on $X$ is $R^{(X)} : = ⨁_{x \in X} R .$ For $x \in X$ , the standard basis vector $e_{x}$ has coordinate $1_{R}$ at $x$ and zero elsewhere. Every element has a unique expression $\sum_{x \in F} r_{x} e_{x}$ with $F \subseteq X$ finite. The map $x \mapsto e_{x}$ is the standard basis inclusion.

More generally, a family $(b_{x})_{x \in X}$ is a basis of a module $M$ when every element of $M$ is uniquely a finite $R$ -linear combination of the $b_{x}$ (Generated submodule, cyclic and finitely generated modules, module basis and free module). For $X = \emptyset$ , $R^{(X)} = 0$ and its empty family is a basis.

Depends on

Used by

ℤ/nℤ is generated but not free as a ℤ-module for n>1 Counterexample
Invariant basis number and the rank of a free module Definition
The standard basis and a universal map from R³ Example
ℤ/2ℤ is projective but not free over ℤ/6ℤ Example
Every injective module is projective (refuted under the Axiom of Choice) False statement
Every projective module is free False statement
Every nonzero commutative ring has invariant basis number for finite bases Theorem
Universal property of the free module on a set Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 6 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)