Remark 1

Let n be an integer $\geq 1$ and S a p-basis of k. One can prove that the subring of $W_n(k)$ generated by $W_n(k^{p^n})$ and by the elements $[\xi, 0, ..., 0]$ ($\xi \in S$), is a p-ring of length n whose residue field is isomorphic to k (cf. p. 72, exerc. 10).