Let $R[x]$ be the polynomial ring over some principal ideal domain $R$. If $R[x]/I$ is free as a $R$-module for some ideal $I$, is $I$ a principal ideal which is generated by some monic polynomial in $R[x]$? Moreover, suppose that $f(x)\in R[x]$ is irreducible and $A=R[x]/(f(x)^i)$ is free as a $R$-module for some $i\geq 1$. Is any $A$-module which is free as a $R$-module isomorphic to $R[x]/((f(x)^j)$ for some $j$ as an $A$-module?

Remark: The above questions are clearly true when $R$ is a field. But I am wondering whether they hold for arbitrary principal ideal domain $R$.

Let $R[x]$ be the polynomial ring over some principal ideal domain $R$. If $R[x]/I$ is free as a $R$-module for some ideal $I$, is $I$ a principal ideal which is generated by some monic polynomial in $R[x]$? Moreover, suppose that $f(x)\in R[x]$ is irreducible and $A=R[x]/(f(x)^i)$ is free as a $R$-module for some $i\geq 1$. Is any indecomposable $A$-module which is free as a $R$-module isomorphic to $R[x]/((f(x)^j)$ for some $j$ as an $A$-module?

Remark: The above questions are clearly true when $R$ is a field. But I am wondering whether they hold for arbitrary principal ideal domain $R$.