Remark

— 1) If $x$ is a nonzero element of $L$ and $c$ a content of $x$, then there exists a unique element $y$ of $L$ such that $x = cy$ ; denote this element $x/c$ ; then $x/c$ is an indivisible element of $L$. 2) The content $c_L(x)$ is the annihilator of the torsion module of $L/Ax$.

Let $L$ be a free module over a principal ideal domain $A$ and let $M$ be a submodule of $L$ ; by VII, p. 4, Lemma 1, the family $c_L(x), x \in M$, has a maximal element ; if $M \neq \{0\}$ then such a maximal element is nonzero.