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.