Does every maximal ideal $M$ in a non-commutative ring $R$ having identity but without zero divisors make $R/M$ a division ring?
The question is equivalent to "Does there exist a non-commutative and simple domain?".
It is a little difficult to construct the example. Thank rschwieb.
No.
There exist noncommutative domains which are simple but which are also not division rings. The most famous is probably the first Weyl algebra.
In any such ring, the zero ideal is maximal, but the quotient is not a division ring.