 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th27:
  nilrad A = sqrt({0.A})
  proof
set N ={a where a is Element of A: ex n being Element of NAT st a|^n in {0.A}};
A1: for a be object st a in nilrad A holds a in sqrt({0.A})
    proof
      let a be object;
      assume a in nilrad A; then
      consider s being nilpotent Element of A such that
A3:   s = a;
      thus thesis by A3, Lm25;
    end;
    for a be object st a in sqrt({0.A}) holds a in nilrad A
    proof
      let a be object;
      assume a in sqrt({0.A}); then
      a in N by IDEAL_1:def 24; then
      consider s being Element of A such that
A7:   s = a & ex n being Element of NAT st s|^n in {0.A};
      consider n be Element of NAT such that
A8:   s|^n in {0.A} by A7;
A9:   s|^n = 0.A by A8,TARSKI:def 1;
      n is non zero Nat by A8,Lm26; then
      s is nilpotent by A9;
      hence thesis by A7;
    end;
    hence thesis by A1,TARSKI:2;
  end;
