by Zorn's lemma:

let be the set of all "proper" right ideals of . (proper means .) this set is not empty because it contains now if is a totally ordered collection of elements of then

is a right ideal of and because if then, since we must have for some which is false. so, by Zorn's lemma, has a "maximal" element, say

it is obvious that is a maximal right ideal of