The problem is trivial if is prime, for then it is maximal and is a field, so that always works for nonzero .

The problem is only hard when has zero divisors. We can suppose is neither a unit nor 0, those cases being trivial.

If are not all distinct then for some we have then works.

If they are all distinct then I don't know. I suspect this is impossible but I can't see why that would be. I'd love the solution!