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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.

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    $\begingroup$ Use the correspondence $\{\text{ideal }J\text{ of }R/I\}$ with $\{\text{ideal }J\supseteq I\text{ of }R\}$. $\endgroup$
    – user10354138
    Commented Feb 20, 2021 at 10:19

1 Answer 1

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Hint:

Use that ideals in a quotient ring $R/I$ correspond bijectively to ideals $J$ of $R$ that contain $I$ by the correspondance $J\longmapsto J/I$. In terms of P.I.D., this means that a generator of the ideal $J$ will divide the generators of $I$.

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