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

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

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

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