Is the following statement true/false?
For a non-trivial ring $R$, and a maximal modular ideal $M\subset R$, if $M$ is not one-sided maximal, then the non-invertible elements of the quotient $R/M$ are zero-divisors.
I know if an ideal is left-maximal or right-maximal, then the quotient must be a division ring. But want to understand why this fails when it's only maximal. Do simple rings/algebras fail to be division rings/algebras only because of zero-divisors? I'm aware of this answer, but that only shows an example. Thanks in advance.
No, it doesn't follow. For example, the Weyl algebra is a ring (with identity) without nonzero zero divisors, and it is also a simple ring. The zero ideal is maximal (and modular of course, as are all ideals in a ring with identity) but not maximal as a left or right ideal.