I am fascinated by matrix rings and their ideals. I am trying to solve a specific problem. First, we recall that any two-sided ideal of $M_n(R)$ is of the form $M_n(I)$ where I is an ideal of the ring $R$, which is commutative and has $1$. Sands in his paper 1 proved that semiprime ideals of $M_n(R)$ is of the form $M_n(I)$, where $I$ is also a semiprime ideal of $R$. My claim is
If every element $x$ of a semiprime ideal $I$ of $R$ is of the form $x=yz$ for some $y,z\in I$, then any element $A\in M_n(I)$ is of the form $A=BC$ for some elements $B,C\in M_n(I)$.
The converse is easy. I need to prove this direction. Can anyone please help?