$p$-torsion of quotient rings Let $A$ be an $p$-adically complete (isomorphic to the inverse limit system) integral domain and $I$ be a prime ideal such that both $A$ and $A/I$ has no $p$-torsion. Is it possible that $A/I^2$ (or higher power $A/I^n$) has $p$-torsion? Basically, I want to show that each $A/I^n$ is $p$-adically complete
 A: Consider the finite field with $p$ elements $\mathbb{F}_{p}$. The kernel $I$ of the canonical map of Witt vectors $W\left(\mathbb{F}_{p}\left[X\right]\right)\to W\left(\mathbb{F}_{p}\right)$ is a prime ideal of the $p$-adically complete integral domain $W\left(\mathbb{F}_{p}\left[X\right]\right)$. The quotient $W\left(\mathbb{F}_{p}\right)$ has no $p$-torsion.
The ideal $I$ consists of Witt vectors $\left(w_{0},w_{1},\ldots\right)\in W\left(\mathbb{F}_{p}\left[X\right]\right)$ such that $X$ divides all $w_{n}$ for $n\in\mathbb{N}$. In particular, the Teichmüller representative $\left[X\right]$ is an element of $I$, as well as the elements $V^{n}\left(\left[X^{p-1}\right]\right)$ and $V^{n}\left(\left[X\right]\right)$ for any $n\in\mathbb{N}^{*}$. This implies that this is an element of $I^{2}$:
\begin{equation*}
V^{n}\left(\left[X^{p-1}\right]\right)V^{n}\left(\left[X\right]\right)=V^{2n}\left(\left[X^{p-1}\right]^{p^{n}}\left[X\right]^{p^{n}}\right)=V^{2n}\left(\left[X^{p^{n+1}}\right]\right)=p^{n+1}V^{n-1}\left(\left[X\right]\right)\text{.}
\end{equation*}
For the second equality, see [Bourbaki, Algèbre commutative, chapitre IX, Proposition 5].
But as $\left[X\right]$ is not an element of $I^{2}$, we get that $W\left(\mathbb{F}_{p}\left[X\right]\right)/I^{2}$ has $p$-torsion.
In fact, $W\left(\mathbb{F}_{p}\left[X\right]\right)/I^{2}$ might not $p$-adically complete. Indeed, according to [The Stacks project, 031A], the ring $A/I^{2}$ is $p$-adically complete if and only if $I^{2}=\bigcap_{n\in\mathbb{N}^{*}}\left(I^{2}+p^{n}A\right)$.
But it is uncertain that series such as $\sum_{n\in\mathbb{N}^{*}}p^{n+1}V^{n-1}\left(\left[X\right]\right)$ are in $I^{2}$, that is it is unclear that it can be written as a finite sum of products of elements of $I$. This seems like a difficult question to me.
Nevertheless, this would have worked if $I$ was principal. Indeed, let $w,g\in W\left(\mathbb{F}_{p}\left[X\right]\right)$. If for all $n\in\mathbb{N}^{*}$ there is $a_{n}\in W\left(\mathbb{F}_{p}\left[X\right]\right)$ such that $w\equiv ga_{n}\pmod{p^{n}}$, then $w=g\left(a_{1}-\sum_{n\in\mathbb{N}^{*}}\left(a_{n}-a_{n+1}\right)\right)$.
In general, $p$-torsion and $p$-completeness are unrelated, as $\mathbb{Z}/p^{2}\mathbb{Z}$ is $p$-complete for instance.
