theorem Th50:
  h is RingHomomorphism implies h"M0 is Ideal of A
   proof
     assume
A1:  h is RingHomomorphism; then
A2:  h"M0 is add-closed by Lm48;
A3:  h"M0 is left-ideal by A1,Lm49;
A6:  dom (h) = the carrier of A by FUNCT_2:def 1;
     h.0.A = 0.B by A1,RING_2:6; then
     h.0.A in M0 by IDEAL_1:2; then
     0.A in h"M0 by A6,FUNCT_1:def 7;
     hence thesis by A2,A3;
   end;
