Ask HN: Complex modulo is a division ring? Can you prove it?

The integers mod n form a mathematical ring. If n is prime, they form a division ring (a ring in which every number has a multiplicative inverse).

The complex integers modulo nr (real) and ni (imaginary) also form a ring. They appear to form a division ring iff 1) nr = ni, and 2) nr is a Mersenne prime.

Does anyone know how to prove that this is in fact a division ring (other than exhaustively)? Can anyone ELI5 (or even ELI20) why it has to be modulo a Mersenne prime?