Proposition 23

In a module $E$ over an integral domain $A$, the set $T$ of non-free elements is a submodule of $E$.

If $x$ and $y$ are not free, there exist two non-zero elements $\alpha, \beta$ in $A$ such that $\alpha x = 0$ and $\beta y = 0$. Then $\alpha \beta \neq 0$ since $A$ is an integral domain and $\alpha \beta (\lambda x + \mu y) = 0$ for all $\lambda$ and $\mu$ in $A$ since $A$ is commutative, hence $\lambda x + \lambda y$ is not free.