A Frobenius map on a commutative ring 
with multiplicative
identity and ring characteristic 
, where 
is a prime number, is the
ring homomorphism 
given by 
. It preserves addition
by the freshman's dream, as well as multiplication
and the multiplicative identity (Stacks
Project, Tag 0CC6).
The Frobenius map is injective iff 
is a reduced
ring. Indeed, its ring kernel consists of the
elements with 
. A nilpotent
element is killed by some iteration of the Frobenius
map, which prevents injectivity unless it is zero.
For a field, the Frobenius map is always injective. It is a field automorphism iff
the field is a perfect field,
and this holds in particular for every finite field
(Stacks Project, Tag 05DU). It need not be surjective
on an infinite field. For example, 
has no 
th root in the field of rational
functions 
, so its Frobenius map is not surjective.
