Remark

Let M and $M'$ be semiprimordial A-modules. It immediately follows from Lemma 4 of VIII, p. 33 that $M'$ is isomorphic to a direct factor submodule of M if and only if we have $[M': L]\leqslant [M : L]$ for every primordial A-module L. In particular, if L is a primordial A-module, [M : L] is the greatest of the cardinals $\mathfrak{a}$ for which there exists a direct factor submodule of M isomorphic to $L^{(\mathfrak{a})}$.

The relation [M : L] = 0 therefore means that there exists no direct factor submodule of M isomorphic to L. This does not exclude the existence of a submodule of M isomorphic to L; it suffices to consider the example where $A =\mathbf{Z}, L =\mathbf{Z}/2\mathbf{Z}$, and $M =\mathbf{Z}/4\mathbf{Z}:$ the $\mathbf{Z}$-modules L and M are primordial and not isomorphic, so that [M : L] = 0 and L is isomorphic to the submodule $2\mathbf{Z}/4\mathbf{Z}$ of M.