
theorem
for R being non degenerated commutative Ring
holds R is reduced iff nilrad R = { 0.R }
proof
let R be non degenerated commutative Ring;
H: nilrad R = the set of all a where a is nilpotent Element of R
   by TOPZARI1:def 13;
A: now assume B: R is reduced;
   C: now let o be object;
      assume o in nilrad R; then
      consider a being nilpotent Element of R such that D: o = a by H;
      a is zero by B;
      hence o in {0.R} by D,TARSKI:def 1;
      end;
   now let o be object;
      assume o in {0.R};
      then o = 0.R by TARSKI:def 1;
      hence o in nilrad R by H;
      end;
   hence nilrad R = { 0.R } by C,TARSKI:2;
   end;
now assume B: nilrad R = { 0.R };
  now let a be Element of R;
    assume a <> 0.R;
    then not a in nilrad R by B,TARSKI:def 1;
    hence a is non nilpotent by H;
    end;
  hence R is reduced;
  end;
hence thesis by A;
end;
