 If "E" is a subset of "R", then formula_55 is a left ideal, called the left ideal generated by "E"; it is the smallest left ideal containing "E".Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of "R".If "x" is in "R", then formula_56 and formula_57 are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by "x".