What are the characteristic submodules of a free $R$-module $F$? Here, $R$ is any ring. By definition, a submodule is characteristic when it is invariant under all automorphisms. I work with left modules.
It is easy to check that all fully-invariant submodules (those which are invariant under all endomorphisms) have the form $IF$ where $I$ is an ideal of $R$. These are in particular also characteristic. When $R$ is a PID, these are all characteristic submodules (see Reference request for characteristic subgroups of free abelian groups - the same argument works here).
Here is an attempt. If $U$ is a characteristic submodule of $F$, and $(e_s)_{s \in S}$ is a basis of $F$, let $I \subseteq R$ denote the ideal generated by all coefficients w.r.t. the basis of the elements in $U$. Clearly, $U \subseteq IF$. The converse holds when $U$ is fully-invariant, but not necessarily when $U$ is only invariant.
As for the case that the rank is one: The characteristic submodules of $R$ are the left ideals $I \subseteq R$ such that $Ir \subseteq I$ for every $r \in R^{\times}$. This differs from an ideal when $R$ is not commutative and there are few units.
Special cases are also appreciated. For example, when $F$ has finite rank, or when $R$ is commutative, Noetherian etc.
I found it also rather surprising how little I could find about the theory of characteristic submodules in the literature. The terminology makes sense for every category, but it seems to be mostly used for the category of groups.
