product of ideals in matrix ring

Question

It well known that if $R$ is a commutative ring then every ideal of $M_n(R)$ is $M_n (I)$ where $I$ is an ideal of $R$. My question is: Suppose that $I$ and $J$ are ideals of $M_n(R)$, i.e., $I=$M_n(A)$ and $J=$M_n(B)$. When is the statement $M_n(AB)=M_n(A)M_n(B)$ correct?

Answer 1

The standard proof of the fact you mention makes use of the fact that if $J$ is an ideal of $M_n(R)$, then $J=M_n(I)$ where $I$ is the set of elements appearing in the upper left hand corner of elements of $J$. That is what establishes the bijective and order preserving correspondence.

For $M_n(A)M_n(B)$, the ideal $I$ is clearly $AB$, so we’re already done.