Hello.

I proofed the following proposition and I am not sure whether there aren't any mistakes:

Let be a principal ideal domain. Then the ideal is irreducible if and only if for a prime ideal and .

( ): If , then for some and for a prime element .

Let , .

. It follow: or and therefore is irreducible.

( ): Let be a irreducible ideal of and be the prime factorisation of with for . Then are coprime. It follows:

.

Let

Because is irreducible, one can assume without loss of generality:

.

, therefore because of .

Now, and both have the factor and therefore are not coprime. This is a contradiction and we get , thus .

Are there any mistakes?

Bye,

Alexander