Corollary 2

Let $C$ be a $p$-ring and $a$ an ideal of $C$ distinct from $C$. The ring $C/a$ is a $p$-ring.

One may suppose $a \neq \{0\}$. There then exists an integer $i \geqslant 1$ such that $a = p^i C$; the ideal $pC/a$ of $C/a$ is maximal and one has $p^i 1_{C/a} = 0$, therefore $C/a$ is a $p$-ring.

Let $C$ be a $p$-ring. The length of $C$, denoted by $l(C)$, is the least upper bound in $\overline{\mathbf{R}}$ of the set of integers $n \geqslant 1$ such that $p^{n-1} 1_C \neq 0$. When $l(C)$ is finite, it is the length of the $C$-module $C$, and when $l(C)$ is equal to $+ \infty$, the $C$-module $C$ is not of finite length (prop. 1).