Lemma 1

Let $L$ be a free module over a principal ideal domain $A$ and let $x$ be an element of $L$. Then the following conditions are equivalent:

(i) $x$ is indivisible ; (ii) *there exists a linear form $f$ on $L$ such that $f(x) = 1$ ; (iii) $x$ is nonzero and the submodule $Ax$ is a direct factor of $L$ ; (iv) $x$ is an element of some basis of $L$.

(i) $\Rightarrow$ (ii) : this follows from the definition. (ii) $\Rightarrow$ (iii) : let $f$ be a linear form on $L$ such that $f(x) = 1$ ; then $x \neq 0$ and the map $y \mapsto f(y)x$ is a projector of $L$, with image $Ax$. (iii) $\Rightarrow$ (iv) : let $L'$ be a complement of $Ax$ in $L$, and let $B'$ be a basis of $L'$ (VII, p. 15, Cor. 2) ; then $B' \cup \{x\}$ is a basis of $L$. (iv) $\Rightarrow$ (i) : trivial.