then "I" is a left ideal if formula_52. Similarly, a right ideal is a subset "I" such that formula_53. A subset "I" is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of "R". If "E" is a subset of "R", then formula_54 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".