TITLE: Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
QUESTION [10 upvotes]: Please help me prove that 

the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain.

This was from Abstract Algebra

REPLY [4 votes]: If the ideal is nontrivial, then the quotient will be a field, a PID; otherwise the quotient will be the original PID.
To prove the first assertion, prove that every nonzero prime ideal in a PID is necessarily maximal.