Remark

(1) By Definition 10, if $A \neq \{0\}$ and $(a_l)_{l \in I}$ is a free family, no element $a_\kappa$ can be equal to a linear combination of the $a_l$ of index $l \neq \kappa$. But conversely, a family $(a_l)$ satisfying this condition is not necessarily a free family. For example, let A be an integral domain and $a, b$ two distinct non-zero elements; in A, considered as an A-module, $a$ and $b$ form a related system, since $(-b)a + ab = 0$. But in general there does not exist an element $x \in A$ such that $a = xb$ of $b = xa$ (cf. however § 7, no. 1, Remark).

An element $x$ of a module E is called free if $\{x\}$ is a free subset, that is if the relation $\alpha x = 0$ implies $\alpha = 0$. Every element of a free subset is free and in particular 0 cannot belong to any free subset when $A \neq \{0\}$.