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 Th34:
    (I <> [#]A &
    for x,y be Element of A st x*y in I & not x in I holds y in sqrt I)
    iff
    A/I is non degenerated &
    (for z be Element of A/I st z is zero_divisible holds z is nilpotent )
    proof
A1:   (I <> [#]A &
      for x,y be Element of A st x*y in I & not x in I holds y in sqrt I)
      implies
      A/I is non degenerated &
      (for z be Element of A/I st z is zero_divisible holds z is nilpotent )
      proof
        assume
A2:     (I <> [#]A &
        for x,y be Element of A st x*y in I & not x in I holds y in sqrt I);
A3:     I is proper by A2;
        (for z be Element of A/I st z is zero_divisible holds z is nilpotent)
        proof
          let z be Element of A/I;
          assume z is zero_divisible; then
          consider y be Element of (A/I) such that
A5:       y <> 0.(A/I) & y*z = 0.(A/I);
          consider z0 being Element of A such that
A6:       z = Class(EqRel(A,I),z0) by RING_1:11;
          consider y0 being Element of A such that
A7:       y = Class(EqRel(A,I),y0) by RING_1:11;
A8:       not y0 in I
          proof
            assume y0 in I; then
A10:        y0 - 0.A in I;
            y = Class(EqRel(A,I),0.A) by A7,A10,RING_1:6
            .= 0.(A/I) by RING_1:def 6;
            hence contradiction by A5;
          end;
          Class(EqRel(A,I),0.A) = 0.(A/I) by RING_1:def 6
          .= Class(EqRel(A,I),y0*z0) by A5,A6,A7,RING_1:14; then
          y0*z0 - 0.A in I by RING_1:6; then
A12:      z0 in sqrt I by A8,A2;
          z0 in {a where a is Element of A:ex n being Element of NAT
          st a|^n in I} by A12,IDEAL_1:def 24; then
          consider z1 be Element of A such that
A13:      z1 = z0 and
A14:      ex n being Element of NAT st z1|^n in I;
          consider n1 be Element of NAT such that
A15:      z1|^n1 in I by A14;
          z0|^n1 - 0.A in I by A15, A13; then
A16:      Class(EqRel(A,I),z0|^n1) = Class(EqRel(A,I),0.A) by RING_1:6
          .= 0.(A/I) by RING_1:def 6;
          n1 <> 0
          proof
            assume n1 = 0; then
A18:        z0|^n1 = 1_A by BINOM:8 .= 1.A;
            not I is proper by A15,A13,A18,IDEAL_1:19;
            hence contradiction by A2;
          end; then
          reconsider n1 as non zero Nat;
          z|^n1 = 0.(A/I) by A16,A6,FIELD_1:2;
          hence thesis;
        end;
        hence thesis by A3;
      end;
      A/I is non degenerated &
      (for z be Element of A/I st z is zero_divisible holds z is nilpotent)
      implies
      (I <> [#]A &
      for x1,y1 be Element of A st x1*y1 in I & not x1 in I holds y1 in sqrt I)
      proof
        assume
A19:    A/I is non degenerated &
        (for z be Element of A/I st z is zero_divisible holds z is nilpotent);
        for x1,y1 be Element of A st x1*y1 in I & not x1 in I
        holds y1 in sqrt I
        proof
          let x1,y1 be Element of A;
          assume
A20:      x1*y1 in I & not x1 in I; then
A21:      y1*x1 - 0.A in I;
reconsider z = Class(EqRel(A,I),x1) as Element of A/I by RING_1:12;
reconsider y = Class(EqRel(A,I),y1) as Element of A/I by RING_1:12;
A22:      z <> 0.(A/I)
          proof
            assume z = 0.(A/I); then
            Class(EqRel(A,I),x1) = Class(EqRel(A,I),0.A) by RING_1:def 6;
            then
            x1 - 0.A in I by RING_1:6;
            hence contradiction by A20;
          end;
          y*z = Class(EqRel(A,I),y1*x1) by RING_1:14 .= Class(EqRel(A,I),0.A)
          by A21,RING_1:6 .= 0.(A/I) by RING_1:def 6; then
          y is zero_divisible by A22; then
          y is nilpotent by A19; then
          consider n being non zero Nat such that
A25:      y|^n = 0.(A/I);
          Class(EqRel(A,I),y1|^n) = 0.(A/I) by A25,FIELD_1:2
          .= Class(EqRel(A,I),0.A) by RING_1:def 6; then
A26:      y1|^n - 0.A in I by RING_1:6;
          reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
          reconsider I as Subset of A;
          y1|^n1 in I by A26; then
          y1 in {a where a is Element of A :
                 ex n be Element of NAT st a|^n in I};
          hence thesis by IDEAL_1:def 24;
        end;
        hence thesis by A19;
      end;
      hence thesis by A1;
    end;
