Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Question

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?

Answer 1

The answer of user39385 to the other question generalizes. The minimal left ideals of $M_n(R)$ are in bijection with the minimal submodules of $R^n$.

Let $K$ be the quotient field of $R$, and let $d$ be the dimension over $K$ of the socle $\operatorname{soc}(R)$ (i.e., the largest semisimple ideal) of $R$. Then a minimal submodule of $R^n$ is just a one-dimensional $K$-subspace of $\operatorname{soc}(R)^n$. If $|K|=q$, then there are $(q^{dn}-1)/(q-1)$ such subspaces.