Remark

(2) A free module can have elements $\neq 0$ which are not free: for example, the A-module $A_s$ is free but the right divisors of zero in A are not free elements of $A_s$.

(3) In the additive group $\mathbf{Z}/(n)$ ($n$ an integer $\geq 2$) considered as a $\mathbf{Z}$-module, no element is free and a fortiori $\mathbf{Z}/(n)$ is not a free module.

(4) It can happen that every element $\neq 0$ of an A-module is free without the module being free. For example, the field $\mathbf{Q}$ of rational numbers is a $\mathbf{Z}$-module with this property, for two elements $\neq 0$ of $\mathbf{Q}$ always form a related system and a basis of $\mathbf{Q}$ could therefore only contain a single element $a$; but the elements of $\mathbf{Q}$ are not all of the form $na$ with $n \in \mathbf{Z}$ (cf. VII, § 3).