Definition 1

Let A be a $k$-algebra and $\mathfrak{a}$ a left ideal of A. An element $u$ of A such that $au-a$ belongs to $\mathfrak{a}$ for every $a\in A$ is called a right unit modulo $\mathfrak{a}$. We say that the ideal $\mathfrak{a}$ is regular if there exists a right unit modulo $\mathfrak{a}$.

Likewise, we say that a right ideal $\mathfrak{b}$ of A is regular if A has a left unit modulo $\mathfrak{b}$, that is, an element $v$ such that $va-a\in \mathfrak{b}$ for every $a\in A$. When an ideal is two-sided, one needs to specify whether it is regular as a left ideal or as a right ideal.

When the algebra A is unital, the unit element of A is a right unit modulo $\mathfrak{a}$ for every left ideal $\mathfrak{a}$ of A; in this case, every left (or right) ideal of A is regular. On the other hand, if A is a $k$-algebra whose elements are all nilpotent, then A is the only (left or right) ideal of A that is regular.

Let $\mathfrak{a}$ be a regular left ideal of A and $u$ a right unit modulo $\mathfrak{a}$. If $\mathfrak{b}$ is a left ideal of A containing $\mathfrak{a}$, then $u$ is a right unit modulo $\mathfrak{a}$, so $\mathfrak{b}$ is regular. So the left ideals of A that are maximal and regular are the maximal elements of the set of regular proper left ideals of A. Moreover, a left ideal $\mathfrak{b}$ of A containing $\mathfrak{a}$ is proper if and only if $u$ does not belong to $\mathfrak{b}$. The set of proper left ideals $\mathfrak{b}$ of A containing $\mathfrak{a}$, ordered by inclusion, is therefore an inductive set. Theorem 2 of Set Theory, III, §2, No. 4, p. 154 applied to this set gives the following result.