Definition 3

— Let $A$ be a commutative ring of characteristic $p \neq 0$. By a perfect closure of $A$ we understand a pair $(A, u)$ where $A$ is a perfect ring of characteristic $p$ and $u$ is a homomorphism of $A$ into $A$ satisfying the following universal property:

(PC) If B is a perfect ring of characteristic p and v is a homomorphism of A into B, then there exists a unique homomorphism h of A into B such that $v = h \circ u$.

The universal property (PC) implies at once the uniqueness of the perfect closure, in the following sense : if $(A, u)$ and $(A', u')$ are two perfect closures of $A$, then there exists a unique isomorphism $h$ of $A$ onto $A'$ such that $u' = h \circ u$ (cf. E, IV, p. 23). We shall now establish the existence of the perfect closure :