 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;
 reserve M0 for Ideal of B;

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;
