Brian Bi

Section 2.4. Ideals

Problem 2.4.1 Let $$I_1 \subseteq I_2 \subseteq \ldots$$ be a chain of proper ideals. Such a chain has an upper bound given by $$I = \bigcup_i I_i$$. It is easy to verify that $$I$$ is an ideal. Since none of the $$I_i$$ contains the unit, neither does $$I$$, so it is also proper. This establishes that every chain of proper ideals has an upper bound which is a proper ideal. By Zorn's lemma, there is a maximal ideal. This holds for left, right, and two-sided ideals.