Basis of a free module

Let $F$ be a free $R$-module . A basis of $F$ is a subset $B\subseteq F$ such that every $x\in F$ can be written uniquely as a finite sum

with coefficients $r_b\in R$, where all but finitely many $r_b$ are zero. Uniqueness is equivalent to the familiar notions of linear independence and spanning (compare linear combinations in linear algebra).

Bases generalize the concept of a basis in a vector space , but over rings bases can fail to exist even for finitely generated modules.

Examples:

• The standard vectors $e_1,\dots,e_n$ form a basis of $R^n$.
• $\{1\}$ is a basis of the free $R$-module $R$.
• (Nonexample) The $\mathbb Z$-module $\mathbb Z/2\mathbb Z$ has no basis, so it is not free.