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

1. ## PID's factor rings have finite amount of principal ideals.

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

Let $R$ be a ring that is a Principal Ideal Domain (Integral domain such that all ideals are principals.), and $A\neq 0$ be an ideal of $R$.

I must show that there is a finite number of ideals of $R/A$, 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 $R/A$ is infinite. If there exists an infinite chain $B_1\subset B_2\subset ...$ of ideals of $R/A$, I can easily show this leads to a contradiction from the fact that $R$ is a PID.

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

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

$Ra \subseteq Rb$ if and only if $a=rb,$ for some $r \in R$. so $b$ is a factor of $a$ and obviously the number of factors of a nonzero element in a PID is finite.