Free module

Definition

Let $R$ be a ring with identity element and let $M$ be a left module over $R$. A subset $E \subseteq M$ is a basis of $M$ when:

1. Each $x \in M$ is a finite sum $r_1 e_1 + \cdots + r_k e_k$ with $r_i \in R$ and $e_i \in E$.
2. If $e_1, \dots, e_k \in E$ are distinct and $r_1 e_1 + \cdots + r_k e_k = 0$, then $r_1 = \cdots = r_k = 0$.

The module $M$ is free when it has a basis.

Remarks

$R^n$ is free, with the standard basis; each vector space is free over its field. For $n \ge 2$, the $\Z$-module $\Z / n\Z$ is not free, because $n x = 0$ for all its elements. Over a nonzero commutative ring, any two bases of a free module have the same cardinality, called the rank. A submodule of a free module need not be free, but over a principal ideal domain it is free.

Explanations

L1 human

A free module is a module that behaves like $R^n$: it has a basis. Every vector space has a basis, but many modules do not. For example, $\Z / 2\Z$ has no basis over $\Z$, because $2x = 0$ for every element $x$.