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:
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?