
theorem
for R being non degenerated commutative Ring holds ker(Frob R) c= nilrad R
proof
let R be non degenerated commutative Ring;
H1: nilrad R = the set of all a where a is nilpotent Element of R
    by TOPZARI1:def 13;
set n = Char R;
R is n-characteristic by RING_3:def 6; then
H2: ker(Frob R) = { a where a is Element of R : a|^n = 0.R } by FrK;
now let o be object;
  assume o in ker(Frob R); then
  consider a being Element of R such that
  A: o = a & a|^n = 0.R by H2;
  now assume n = 0;
    then a|^n = 1_R by BINOM:8;
    hence contradiction by A;
    end;
  then a is nilpotent by A,TOPZARI1:def 2;
  hence o in nilrad R by A,H1;
  end;
hence thesis;
end;
