that is good that you are unsure how to go the other way because the other direction is simply not true. It is only true if your ring is a principal ideal domain.

For a counterexample consider polynomials over the integers. Consider the ideal (x).

which is an integral domain and is in particular not a field. The fact that it is an integral domain shows it is prime, and the fact that it is not a field shows that it is not maximal. So there is an example of a prime ideal which is not maximal in a commutative ring with 1.