Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring $A$ (not necessarily commutative), the matrix ring $M_n(A)$ is simple if and only if $A$ is simple.
How to prove?
All ideals below are two-sided ideals. I also assume $A$ has a unit.
If $I$ is an ideal of $A$, then $J=M_n(I)$ is an ideal of $M_n(A)$.
This is easy to prove from the definition of matrix addition and multiplication.
If $J=M_n(I)$ is an ideal of $M_n(A)$, then $J=M_n(I)$ for some $I$ is an ideal of $A$.
Take $I$ to be the set of all $a \in A$ such that there is $M\in J$ having $a$ in one entry. Then $I$ is an ideal of $A$ and $J=M_n(I)$ because you can pre- and post- multiply matrices in $J$ by elementary matrices $E_{ij}$ to move entries to any desired place. ($E_{ij}$ is the matrix will all zeros except for the entry $ij$ which is $1$.)