Let $R$ be a PID. I have to show that every ideal of a quotient ring of $R$ is a principal ideal. I am able to visualize this problem by taking $R=Z$. I took ideal $(20)$ of $Z$ for example. Then I got quotient ring as: $R/(20)=\{(20),1+(20),2+(20),.......,19+(20)\}$. When I consider an ideal of this quotient ring e.g. $\{(20),2+(20),4+(20),.....,18+(20)\}$,
I can see that this ideal is generated by a single element of the ideal i.e. $2+(20)$. Thus it is a principal ideal. Similar thing can be said about other ideals of the quotient ring. But when it comes to formal proof of this, I do not have a clue as to how to do that. Please suggest.