Remark

Let $E$ be a module over any commutative ring $A$. If $x$ is a non-free element of $E$, every element of the submodule $Ax$ is non-free. On the other hand, if $A$ contains divisors of 0, the sum of two non-free elements of $E$ may be free; for example, in $\mathbf{Z}/6\mathbf{Z}$ considered as a module over itself, 3 and 4 are not free, but $3 + 4 = 1$ is free.

Proposition 23 leads to the following definition: