Remark

The proof of Th. 1 shows that every submodule of $A^{(I)}$ is isomorphic to a direct sum $\bigoplus a_i$, where each $a_i$ is an ideal of $A$.

PROPOSITION I. — If L is a free module of finite rank n over a principal ideal domain A, then every submodule M of L is a free module of rank $\leq n$. Indeed M is a free module by Cor. 2 to Th. 1, and it has rank $\leq n$ by the previous remark, or by the following lemma: