Show that $([A,A])$ contains the identity matrix.

Question

Suppose $A = M_n(C)$, $n \geq 1$, where $C$ is a commutative unital ring. If $n \geq 2$, then $([A,A])$, the ideal generated by $[A,A]$, contains the identity matrix.

Answer 1

(Revised) Step 1:It is easy to prove that $E_{ij}\in [A,A]$ for all $i\neq j$. Also $E_{ii}-E_{jj}\in [A,A]$.

Step 2: Using step 1, let's prove that $E_{ii}\in ([A,A])$.

Step 3: Using step 2, it is easy to prove that $I\in ([A,A])$.

Just please use different $E_{ij}$ to obtain all of these..Also we have two sided ideals.:)