On a property of nilpotent matrices over an algebraically closed field

Suppose $F$ is an algebraically closed field. We prove that the ring $\prod_{n=1}^\infty \mathbb M_n(F)$ has a special property which is, somewhat, in sharp parallel with (and slightly better than) a property established by Šter (LAA, 2018) for the rings $\prod_{n=1}^\infty \mathbb M_n(\mathbb Z_2)$ and $\prod_{n=1}^\infty \mathbb M_n(\mathbb Z_4)$, where $\mathbb Z_2$ is the finite simple field of two elements and $\mathbb Z_4$ is the finite indecomposable ring of four elements.