Example 2

Let $K$ be a perfect field of characteristic $p$. By prop. 8 of § 1, No. 8, the ring $W(K)$ of Witt vectors is a $p$-ring of infinite length. The mapping $(a_n)_{n \in \mathbf{N}} \mapsto a_0$ induces, by passing to quotients, an isomorphism of $W(K)/pW(K)$ onto the field $K$ (loc. cit., prop. 7). For every integer $n \geqslant 1$, the ring

is a $p$-ring of length $n$.