TITLE: Let $a$ be a non-unit in a ring $R$. Show that $a$ lies in a maximal ideal.
QUESTION [5 upvotes]: Let $a$ be a non-unit in a ring $R$. Show that $a$ lies in a maximal ideal.
Is there a way to solve this without using Zorn's Lemma?

REPLY [5 votes]: No; without Zorn's lemma, a ring may not have any maximal ideal. See Zorn's lemma in abstract algebra?.