I have to proof, that is a noetherian ring and a faithfully flat -algebra.
Furthermore, I have to proof, that for every ideal in the ring is a Artinian ring und every prime ideal in is a maximal ideal in .
Would you please check my proof? :
is a Euclidean domain, hence it is principal ideal domain and therefore noetherian, because every principal ideal domain is a noetherian ring.
Or another idea: is isomorphic to as -module. The -module is noetherian (e.g. is noetherian ring) and therefore is also a noetherian -module.
faithfully flat -algebra:
Let a exact sequence of -moduls. Then
is exact and we have
is exact is exact, this means that is a faithfully flat -algebra.
Now the other part:
There is a one-to-one correspondence between the ideals in and the ideals in with .
is a principal ideal domain, this means, that for every ideal in there exists an element with .
There is only a finite number of divisors of a. It follows, that there is only a finite number of ideals in with , thus there is only a finite number of ideals in the ring and so is a artinian ring.
It follows: Every prime ideal in is a maximal ideal in , hence every prime ideal in is a maximal ideal in .
Are there any mistakes in my proof? I would be glad if someone would check the proof.
Thanks in advance!