reserve R for commutative Ring;
reserve A for non degenerated commutative Ring;
reserve I,J,q for Ideal of A;
reserve p for prime Ideal of A;
reserve M,M1,M2 for Ideal of A/q;

theorem Th39:
    for q be proper Ideal of A holds sqrt q is maximal implies q is primary
    proof
      let q be proper Ideal of A;
      set m = sqrt q;
      assume
A1:   m is maximal; then
A2:   (canHom q).:m is maximal by TOPZARI1:20,Th30;
      reconsider  M = (canHom q).:m as Ideal of A/q by Th19;
A3:   A/q is local by A1,Th38;
      for x be Element of A/q st x is zero_divisible holds x is nilpotent
      proof
        let x be Element of A/q;
        assume
A4:     x is zero_divisible;
        x is NonUnit of A/q
        proof
          assume not x is NonUnit of A/q; then
          x in {a where a is Element of A/q: a is Unit of A/q }; then
          x in Unit_Set(A/q) by RINGFRAC:def 5; then
          consider a1 be Element of A/q such that
A6:       x*a1 = 1.(A/q) by Lm3;
          consider y be Element of A/q such that
A7:       y <> 0.(A/q) & y*x = 0.(A/q) by A4;
          y  = y*1.(A/q) .= (y*x)*a1 by A6,GROUP_1:def 3 .= 0.(A/q) by A7;
          hence contradiction by A7;
        end; then
        consider m1 be maximal Ideal of A/q such that
A8:     x in m1 by TOPZARI1:10;
   set S = {I where I is Ideal of A/q: I is quasi-maximal & I <> [#](A/q)};
        M in S by A2; then
A9:     M in m-Spectrum (A/q) by TOPZARI1:def 7;
        m1 in S; then
A10:    m1 in m-Spectrum (A/q) by TOPZARI1:def 7;
        consider M0 be Ideal of A/q such that
A11:    m-Spectrum (A/q) = {M0} by A3;
        M = M0 by A9,A11,TARSKI:def 1 .= m1 by A10,A11,TARSKI:def 1; then
        x in nilrad(A/q) by Th37,A8; then
        x in the set of all a where a is nilpotent Element of A/q
          by TOPZARI1:def 13; then
        consider x0 be nilpotent Element of A/q such that
A13:    x0 = x;
        thus thesis by A13;
      end;
      hence thesis by Th35;
    end;
