Let $A$ be subring of $\text{Mat}_n(Z_m)$. Suppose, for $x\in A$, $x^2=0$ implies $x=0$.
Claim A is commutative.
Attempt
$A$ is finite ring, hence Artinian.
If it is possible to claim that Jacobson Radical is $0$,
then by Weddenburg-Artin, result might follow.
(I did not finish the argument yet, I am not sure if this path is okay.)
I am not sure how to use the information $x^2=0 \implies x=0$.
Also, I am not sure where to use being subring of $\text{Mat}_n(Z_m)$ (except being finite).
A brief explanation might help a lot, thanks in advance.