Rank of a free module

Let $F$ be a free module . The rank of $F$ is the cardinality of a basis , denoted $\operatorname{rank}(F)$. This is well-defined: any two bases of a free module have the same cardinality.

When the rank is finite, it plays the role of dimension , but over a general ring one typically speaks of rank only for free (or locally free) modules.

Examples:

• $\operatorname{rank}(R^n)=n$.
• The zero module has rank $0$ (its basis is the empty set).
• If $F\cong \bigoplus_{i\in I} R$, then $\operatorname{rank}(F)=|I|$.