Rank of a free module

Let $R$ be a nonzero commutative ring and $F$ a free $R$ -module. The rank of $F$ is the cardinality of a basis, denoted $\operatorname{rank}(F)$ . This is well-defined because any two bases of a free module over a nonzero commutative ring 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 $\operatorname{rank}(F)=|I|$ .