Proposition 6

Let C be a p-ring, whose residue field k be perfect. Suppose that C is of finite length n (resp. infinite). There exists a unique isomorphism $u : W_n(k) \to C$ (resp. $u : W(k) \to C$) which induces, by passing to quotients, the identical mapping of k.

Since $W_n(k)$ (resp. $W(k)$) is a p-ring with residue field k, and of length n (resp. of infinite length) (No. 1, Example 2), and since $\varnothing$ is a p-basis of the perfect field k, prop. 6 is a particular case of prop. 4 of No. 3.