Let be a semigroup and let . Let , , and be the left, right, and two-sided ideals generated by , respectively. Prove that , , and .
Recall that by definition, is the intersection over the class of left ideals which contain . Suppose is a left ideal containing . Then for all , , and thus . So . On the other hand, we have . So is a left ideal containing , and thus . So .
The proofs for and are similar; show that every (right,two sided) ideal contains the set in question, and that the set in question is a (right, two-sided) ideal.