\[ \DeclareMathOperator{\End}{End} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\span}{span} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\ad}{ad} \newcommand\d{\mathrm{d}} \newcommand\pref[1]{(\ref{#1})} \]

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.