
theorem
for R being Ring,
    I being Subset of R
holds I is Ideal of R iff
      ex S being R-homomorphic Ring,
         f being Homomorphism of R,S st ker f = I
proof
let R be Ring,
    I be Subset of R;
now assume A: I is Ideal of R;
  thus ex S being R-homomorphic Ring,
          f being Homomorphism of R,S
       st ker f = I
    proof
    reconsider I as Ideal of R by A;
    take R/I,canHom I;
    thus thesis by kercanhomI;
    end;
  end;
hence thesis;
end;
