Free module

A left $R$-module $F$ is free if it has a basis $B\subseteq F$, i.e. every element of $F$ admits a unique expression as a finite $R$-linear combination of elements of $B$. Equivalently, $F$ is isomorphic to a direct sum of copies of $R$ indexed by $B$.

Free modules are characterized by the universal property of free modules : functions from a basis extend uniquely to module homomorphisms. Over a field, free modules coincide with vector spaces and their bases agree with linear-algebra bases.

Examples:

• $R^n$ is a free $R$-module with basis $\{e_1,\dots,e_n\}$.
• $\mathbb Z^n$ is a free $\mathbb Z$-module of rank $n$.
• (Nonexample) $\mathbb Z/2\mathbb Z$ is not free as a $\mathbb Z$-module (it has torsion).