Corollary 2

Let V be a left vector space over D and $\mathscr{F}$ a set of endomorphisms of V. An A-submodule of $S\otimes_DV$ is stable under all endomorphisms $1_S\otimes u$, where $u$ runs through $\mathscr{F}$, if and only if it is of the form $S\otimes_DW$, where W is a linear subspace of V that is stable under all endomorphisms belonging to $\mathscr{F}$.

Indeed, by Theorem 2, every A-submodule N of $S\otimes_DV$ is equal to $S\otimes_DW$, where W is the linear subspace of V consisting of the elements $v$ such that $s\otimes v$ belongs to N for every $s\in S$.