Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that $R=\textrm{End}_R(X)$, consider the contravariant functor $h_X:\mathcal{M}\to\mathcal{M}$ given by $T\to\textrm{Hom}_R(T,X)$, where the structure of $R$-module on $h_X(T)$ is induced from that on $X$.

I am interested in having conditions on the ring $R$ ensuring the existence of an $X$ as above for which $h_X$ is an anti-equivalence of categories.

So far I only know that in the following two cases such an $X$ exists:

- if $R$ is the maximal order of $K$, then $X=R$ works;
- if $K$ has a subfield $K_0$ with $[K:K_0]=2$ and such that $R\cap K_0$ is the maximal order of $K_0$ then $X=R$ works.

(The question is closely related to (and inspired by) the following: let $k$ be a finite field and $\mathcal{C}$ a $k$-isogeny class of $k$-simple ordinary abelian varieties over $k$. Does there exist an object $A$ of $\mathcal{C}$ such that any finite $k$-subgroup $H\subset A$ is equal to the intersection of the kernels of a suitable (finite) collection of isogenies $\varphi_i:A\to A$?)