[proofplan]
We prove the equivalence in two directions. The forward direction verifies that a complete DVR of mixed characteristic with uniformizer $p$ and perfect residue field satisfies the three axioms of a strict $p$-ring ($p$-torsion free, $p$-adically complete, perfect residue ring) with $A/pA$ a field. The reverse direction shows that a strict $p$-ring $A$ with $A/pA$ a field is a DVR with unique prime $p$, by proving that every element outside $pA$ is a unit via the geometric series.
[/proofplan]
[step:Forward direction: a complete DVR with uniformizer $p$ and perfect residue field is a strict $p$-ring]
Assume $A$ is a complete DVR of mixed characteristic with uniformizer $p$ and perfect residue field $k = A/pA$. We verify the three defining properties of a strict $p$-ring:
1. **$p$-torsion free.** Since $A$ is an integral domain (it is a DVR, hence in particular a domain) and $p \neq 0$ in $A$ (as $A$ has mixed characteristic, meaning $\operatorname{char}(A) = 0$), the element $p$ is a non-zero-divisor. So $pa = 0$ implies $a = 0$.
2. **$p$-adically complete.** Since $p$ is the uniformizer, the $p$-adic topology coincides with the $\mathfrak{m}_A$-adic topology (as $\mathfrak{m}_A = pA$). A complete DVR is by definition complete with respect to the $\mathfrak{m}_A$-adic topology, so $A$ is $p$-adically complete.
3. **$A/pA$ is perfect of characteristic $p$.** Since $A$ has mixed characteristic, $\operatorname{char}(A/pA) = p$. The residue field $k = A/pA$ is perfect by hypothesis.
Moreover, $A/pA = k$ is a field by definition of a DVR (the residue ring modulo the unique maximal ideal is a field).
[/step]
[step:Reverse direction: show every element outside $pA$ is a unit]
Assume $A$ is a strict $p$-ring with $A/pA = k$ a field. Let $x \in A \setminus pA$. We show $x \in A^\times$.
By the [Witt Expansion in Strict $p$-Rings](/theorems/???), write $x = \sum_{n=0}^\infty [x_n] p^n$ with $x_n \in k$. Since $x \notin pA$, the zeroth coefficient satisfies $x_0 \neq 0$ in $k$ (otherwise $x \equiv 0 \pmod{p}$, contradicting $x \notin pA$).
Since $k$ is a field, $x_0^{-1}$ exists in $k$. The [Teichmüller Map](/theorems/???) gives $[x_0^{-1}] \in A$ with $[x_0^{-1}] \cdot [x_0] = [x_0^{-1} x_0] = [1] = 1$ (using multiplicativity of the Teichmüller map). Multiplying $x$ by $[x_0^{-1}]$:
\begin{align*}
[x_0^{-1}] \cdot x = [x_0^{-1}] [x_0] + p(\text{higher terms}) = 1 + py
\end{align*}
for some $y \in A$ (where $y$ collects all the higher-order terms).
The element $1 + py$ is invertible via the geometric series: since $|py| \leq |p| < 1$ (in the $p$-adic topology), the series $\sum_{m=0}^\infty (-py)^m$ converges in $A$ by $p$-adic completeness, and $(1 + py) \sum_{m=0}^\infty (-py)^m = 1$. Therefore $[x_0^{-1}] \cdot x$ is a unit, and since $[x_0^{-1}]$ is also a unit (with inverse $[x_0]$), $x$ is a unit.
[guided]
The geometric series argument is the $p$-adic analogue of the real-analysis fact that $|r| < 1$ implies $\sum r^n = 1/(1-r)$.
We need to verify convergence: the partial sums $s_N = \sum_{m=0}^{N} (-py)^m$ form a Cauchy sequence because $|(-py)^m| \leq |p|^m \to 0$. The $p$-adic completeness of $A$ then provides a limit $s = \sum_{m=0}^\infty (-py)^m \in A$. That $(1 + py) \cdot s = 1$ follows from telescoping: $(1 + py)(1 - py + (py)^2 - \cdots + (-py)^N) = 1 - (-py)^{N+1} = 1 + (-1)^N (py)^{N+1}$, and the error term $(py)^{N+1} \to 0$ as $N \to \infty$.
The conclusion $A \setminus pA = A^\times$ means that $pA$ is the unique maximal ideal, every nonzero ideal is generated by a power of $p$ (since every nonzero element $x$ has a unique representation $x = p^n u$ with $u \in A^\times$ and $n = $ the $p$-adic valuation of $x$), and $A$ is a DVR with uniformizer $p$.
[/guided]
[/step]
[step:Conclude that $A$ is a complete DVR of mixed characteristic with uniformizer $p$]
From the previous step, $A^\times = A \setminus pA$ and $pA$ is the unique maximal ideal. Every nonzero element $x \in A$ can be written as $x = p^n u$ where $n \geq 0$ is determined by the Witt expansion (specifically, $n$ is the smallest index with $x_n \neq 0$) and $u \in A^\times$. The ideals of $A$ are therefore $p^n A$ for $n \geq 0$, so $A$ is a PID with unique prime $p$ up to units -- that is, $A$ is a DVR with uniformizer $p$.
For completeness: $A$ is $p$-adically complete by the strict $p$-ring hypothesis, and the $p$-adic topology coincides with the $\mathfrak{m}_A$-adic topology since $\mathfrak{m}_A = pA$. So $A$ is a complete DVR.
For mixed characteristic: $A$ contains $\mathbb{Z}$ (since $A$ is $p$-torsion free, the natural map $\mathbb{Z} \to A$ is injective), so $\operatorname{char}(A) = 0$, while $\operatorname{char}(A/pA) = p$. This is the definition of mixed characteristic.
[/step]
