Brian Bi
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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.