
theorem FrK:
for n being Nat
for R being n-characteristic Ring
holds ker(Frob R) = { a where a is Element of R : a|^n = 0.R }
proof
let p be Nat, R be p-characteristic Ring;
set f = Frob R;
H: ker f = {v where v is Element of R : f.v = 0.R} by VECTSP10:def 9;
I: Char R = p by RING_3:def 6;
A: now let o be object;
   assume o in ker(Frob R);
   then consider a being Element of R such that
   B: o = a & f.a = 0.R by H;
   f.a = a|^p by I,defFr;
   hence o in { a where a is Element of R : a|^p = 0.R } by B;
   end;
now let o be object;
  assume o in { a where a is Element of R : a|^p = 0.R };
  then consider a being Element of R such that
  B: o = a & a|^p = 0.R;
  f.a = 0.R by I,B,defFr;
  hence o in ker f by B,H;
  end;
hence thesis by A,TARSKI:2;
end;
