Full statement and proof
Theorem. Let be a ring, and let be a (left, right, two-sided) ideal of that is distinct from . Then there exists a maximal (left, right, two-sided) ideal of containing .
Proof. Note that an ideal of is distinct from if and only if it does not contain 1. Let be the family of proper ideals of containig . Evidently, 1 is not an element of any member of this family, so the union of a totally ordered subset of this family is an element of the family. It then follows from Zorn's Lemma that has a maximal element.