Thread: PID's factor rings have finite amount of principal ideals.

Hey, I have a quick question about Principal Ideal Domains and Ideals.

Let be a ring that is a Principal Ideal Domain (Integral domain such that all ideals are principals.), and be an ideal of .

I must show that there is a finite number of ideals of , and that they are all principal.

Showing they are principal is quite easy. However I'm having problems with showing there is a finite number of them.

What I've done so far;

Suppose the number of distinct ideals of is infinite. If there exists an infinite chain of ideals of , I can easily show this leads to a contradiction from the fact that is a PID.

It seems to me that an infinity of ideals would necessarily imply an infinite chain (intuitively), since any ideal of is going to be of the form , where is contained in , but I can't find a way to either prove of disprove this claim.

if and only if for some . so is a factor of and obviously the number of factors of a nonzero element in a PID is finite.