We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length.
Let $M \neq 0$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring over a division ring $D$. We denote $D$ by $D(R)$. We denote the set of $R$-submodules of $M$ by $L(M)$. We regard $L(M)$ as an ordered set by the inclusion relation.
Let $N \neq 0$ be a left module over a ring $S$. Suppose length $M$ = length $N$ and $D(R)$ is isomorphic to $D(S)$. Is $L(M)$ isomorphic to $L(N)$?