1. The problem statement, all variables and given/known data Given a commutative ring with unity, show that if every ideal is prime than the ring is a field. 2. Relevant equations 3. The attempt at a solution I think that I can show that a ring is a field iff it has no nontrivial ideals. So I guess I need to show that if a ring has only prime ideals than these ideals must be trivial. I'm not sure how to do this though.