Hi,

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-onecorrespondencebetween 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 .

Let .

.

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!

Bye,

Lisa