Division of ideals

Hi,

I have a couple of questions concerning algebraic number theory. Why does I contains J imply I divides J? Firstly assuming I contains J how do we know there exists a fractional ideal I' such that I'I=J and then how do we get that I' is actually an integral ideal?

Another problem I am having is in proving: if I is ideal and N(I)=p some prime p then I is prime ideal. So we assume A,B are ideals such that AB is contained in I. We want to show either A is contained in I or B is. Assuming the above then we know I divides AB. The proof then states that I=ABC some ideal C. I thought I divides AB means IC=AB for some ideal C not other way round. I get that we could multiply by C inverse to get I on its own but then C inverse is a fractional ideal.

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Reply:

I’m presuming that you’ve read the course notes at least twice or so at this point. So for your first question, it’s already assumed that we’ve shown that the non-zero fractional ideals form a group. This gives the existence of the I' you mention. But then
I'I\subset J\subset I, so each element of I' stabilizes the ideal I. This implies that every element of I' is an algebraic integer, i.e., that I' is an ideal.

For the statement about prime ideals, note that we are assuming the factorization theorem. So any I can be written

I=\prod P_i

for prime ideals P_i. But then

N(I)=\prod N(P_i).

Therefore, if N(I) is prime, only one P_i can occur.

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