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