Definition 2

— A ring $A$ of characteristic $p \neq 0$ is said to be perfect if it is commutative and the mapping $a \mapsto a^p$ is bijective.

If the ring $A$ is perfect of characteristic $p$, the mapping $a \mapsto a^{p^f}$ is an automorphism of the ring $A$ for every integer $f \geq 0$; the inverse automorphism is denoted by $a \mapsto a^{1/p^f}$ or $a \mapsto a^{p^{-f}}$ and the image of a subset $S$ of $A$ under this automorphism is written $S^{1/p^f}$ or $S^{p^{-f}}$. It is clear that $(a^{p^e})^{p^f} = a^{p^{e+f}}$ for all $a \in A$ and any integers $e$ and $f$ (of whatever sign).

Let $A$ be a commutative ring of characteristic $p$. For every integer $f \geq 0$ let us write $\mathfrak{n}_f$ for the kernel of the endomorphism $a \mapsto a^p$ of the ring $A$. Then $(\mathfrak{n}_f)_{f \geq 0}$ is an increasing sequence of ideals of $A$; since every positive integer is majorized by a power of $p$, the ideal $\mathfrak{n} = \bigcup_{f \geq 0} \mathfrak{n}_f$ consists of all the nilpotent elements of $A$. In particular, if $A$ is perfect, every nilpotent element of $A$ is zero.