
theorem div0:
for R being comRing,
    a,b being Element of R
holds a divides b iff b in {a}-Ideal
proof
let R be comRing, a,b be Element of R;
A: now assume a divides b;
   then consider c being Element of R such that A1: a * c = b;
   b in the set of all a*r where r is Element of R by A1;
   hence b in {a}-Ideal by IDEAL_1:64;
   end;
now assume b in {a}-Ideal;
  then b in the set of all a*r where r is Element of R by IDEAL_1:64;
  then consider c being Element of R such that A1: a * c = b;
  thus a divides b by A1;
  end;
hence thesis by A;
end;
