Prove $R/M$ is a division ring for a non-commutative ring $R$ with max ideal $M$
I am trying to prove that $R/M$ is a division ring if $M$ is a maximal ideal of a non-commutative ring $R$. I tried by using similar arguments as in $R$ commutative. But the proof that $R/M$ is a field if $M$ is maximal fails if $R$ is non-commutative.
This isn't true. For instance, if $k$ is a division ring, then the ring $M_n(k)$ of $n\times n$ matrices over $k$ has no nontrivial two-sided ideals, so $0$ is a maximal ideal in this ring. But if $n>1$, then $M_n(k)/0\cong M_n(k)$ is not a division ring.