Corollary

Let C be a p-ring of finite length n. There exists a p-ring $C'$ of infinite length such that C is isomorphic to $C'/p^nC'$.

By prop. 5, there exists a p-ring $C'$ of infinite length such that $\kappa_{C'}$ is isomorphic to $\kappa_C$. Then $C'/p^nC' = C'_n$ is a p-ring of length n, and the field $\kappa_{C'_n}$ is isomorphic to $\kappa_{C'}$, hence to $\kappa_C$. By prop. 4, the rings C and $C'_n$ are therefore isomorphic.