Let $A$ be a (not necessarily unital)commutative ring. We define a regular ideal $I$ if exists element $e \in A$, such that $e$ is the unit in $A/I$.
Can we deduce, that every proper regular ideal is contained in a maximal regular ideal?
It seems that this is another Zorn's lemma argument: let $m_i$ be a chain of proper, regular ideals then $\bigcup m_i$ is a proper ideal, but is it regular?