Corollary 1

If the $p$-ring $C$ is an integral domain, it is a discrete valuation ring, or a field of characteristic $p$.

Suppose $C$ is an integral domain. If $p 1_C$ is nilpotent, one has $p 1_C = 0$, and $\{0\}$ is a maximal ideal of $C$, hence $C$ is a field of characteristic $p$. If $p 1_C$ is not nilpotent, then $C$ is a discrete valuation ring by Prop. 1, c).