Perfect ring of characteristic p ($a\mapsto a^p\text{ is bijective}$)

A commutative ring of prime characteristic $p$ is perfect in the Frobenius sense when its Frobenius endomorphism is an automorphism. It is necessarily reduced: if $a$ is nilpotent, some $p$-power iterate is zero, and injectivity gives $a=0$. This use of perfect is distinct from the homological notion of a perfect ring. The Absolute Frobenius morphism of a scheme is an isomorphism exactly when all its local rings are perfect in this sense, because it is the identity on points and is an isomorphism on every stalk.