Prove that a quotient of a principal ideal domain by a prime ideal is again a principal ideal domain.

Let be a principal ideal domain and let be a prime ideal. Note that is an integral domain.

Now by the lattice isomorphism theorem for rings, every ideal of has the form , where is an ideal of containing . Since is a principal ideal domain, for some . Then is principal, and thus every ideal of is principal.

Note that every ideal of a quotient of is principal; we needed to be prime so that is also an integral domain.

## Comments

How about this. Assume I is non-zero. Since I is prime and R is a PID, I is maximal and so R/I is a field and hence a PID.

That works too.