 Let "R" be a ring.A nonempty subset "I" of "R" is then said to be a left ideal in "R" if, for any "x", "y" in "I" and "r" in "R", formula_42 and formula_50 are in "I".If formula_51 denotes the span of "I" over "R", that is, the set of finite sums
then "I" is a left ideal if formula_53.