Dedekind domains with finitely many primes are PIDs
A commutative ring in which there are only finitely many maximal ideals is known as a semi-local ring. For such rings, the property of being a Dedekind domain
and of being a principal ideal domain
coincide.
Theorem.
A Dedekind domain in which there are only finitely many prime ideals is a principal ideal domain.
This result is sometimes proven using the chinese remainder theorem or, alternatively, it follows directly from the fact that invertible ideals in semi-local rings are principal.
Suppose that is a Dedekind domain such as the ring of algebraic integers in a number field. Although there are infinitely many prime ideals in such a ring, we can use the result that localizations of Dedekind domains are Dedekind and apply the above theorem to localizations of .
In particular, if is a nonzero prime ideal, then is a Dedekind domain with a unique nonzero prime ideal, so the theorem shows that it is a principal ideal domain.