
theorem
 for R being Ring holds R/{0.R}, R are_isomorphic
proof
let R be Ring;
id R is RingHomomorphism;
then reconsider S = R as R-homomorphic Ring by defhom;
reconsider f = id R as Homomorphism of R,S;
A: ker f = {0.R} by ker0;
set B = R/(ker f);
B: rng canHom f = the carrier of Image f by FUNCT_2:def 3
               .= rng f by defim;
then reconsider g  = canHom f as Function of B,S by FUNCT_2:6;
C1: now let x,y be Element of B;
    thus g.(x+y)
      = (canHom f).x + (canHom f).y by VECTSP_1:def 20
     .= ((the addF of S)||(rng f)).((canHom f).x,(canHom f).y) by defim
     .= g.x + g.y;
    end;
C2: now let x,y be Element of B;
    thus g.(x*y)
      = (canHom f).x * (canHom f).y by GROUP_6:def 6
     .= ((the multF of S)||(rng f)).((canHom f).x,(canHom f).y) by defim
     .= g.x * g.y;
    end;
g.(1.B) = (canHom f).(1_B)
       .= 1_(Image f) by GROUP_1:def 13 .= 1.S by defim; then
C: g is additive multiplicative unity-preserving by C1,C2;
g is onto by B,FUNCT_2:def 3;
hence thesis by A,C;
end;
