I am aware of the lemma that every regular left ideal of a ring is contained in a maximal left ideal that is regular. But still things are not very clear. Any help will be appreciated.
2026-03-29 13:23:02.1774790582
Is there any difference between a maximal regular left ideal and a regular maximal left ideal?
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Let $\mathcal{A}$ be the subset of maximal left ideals which are regular. Let $\mathcal{B}$ be the set of elements which are maximal in the poset of (proper) regular left ideals.
Obviously $\mathcal{A}\subseteq\mathcal{B}$, but the question is whether or not the reverse containment holds.
From the definition, we know that any left ideal containing a regular left ideal is automatically regular. So an element of $\mathcal{B}$ cannot be properly contained in a maximal left ideal, because that would also have to be regular as well. So yes, $\mathcal B\subseteq \mathcal A$ as well.