Morita equivalence and lattice of subrings

If $R$ and $S$ are two Morita equivalent rings, it is well known that their ideal lattices $\mathfrak{L}(R)$ and $\mathfrak{L}(S)$ are isomorphic. Namely, if we have a progenerator $(R,S)$-bimodule $P$, then an ideal $I$ of $R$ corresponds to the ideal $J$ of $S$ such that $IP=PJ$.

I know that, in general, there is no relation between the lattice of subrings of $R$ and $S$ that we can obtain from the Morita equivalence of $R$ and $S$, but in some special cases, something can be said.

Suppose, for instance, that $R$ is a ring such that every finitely generated projective module is free (polynomial rings over a field, (noncommutative) local rings, left principal ideal domains, etc). Then $S \simeq M_n(R)$ for some $n \in \mathbb{N}$, and in particular the lattice of subrings of $R$ injects into the lattice of subrings of $S$: a subring $A$ of $R$ is sent to $M_n(A)$.

What are other interesting cases where something can be said about the lattice of subrings given a Morita equivalence?

In particular, I am interested in the following question: let $R$ and $S$ be two Morita equivalent rings. Then, as is well known, $S \simeq e M_n(R) e$, for some matrix ring $M_n(R)$ over $R$ and some full idempotent $e \in M_n(R)$ (i. e., $M_n(R)eM_n(R)=M_n(R)$).

Given $A$ a subring of $R$, are there reasonable conditions that make us able to choose the idempotent $e$ as an element in $M_n(A)$?