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.
A Dedekind domain in which there are only finitely many prime ideals is a principal ideal domain.
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.
|Title||Dedekind domains with finitely many primes are PIDs|
|Date of creation||2013-03-22 18:35:18|
|Last modified on||2013-03-22 18:35:18|
|Last modified by||gel (22282)|