Proposition 2

Let A be a commutative $k$-algebra. An ideal $\mathfrak{a}$ of A is a regular maximal ideal if and only if the pseudoring $A/\mathfrak{a}$ (I, §8, No. 1, p. 98) is a field.

An element $u$ of A is a (right or left) unit modulo $\mathfrak{a}$ if and only if the canonical image of $u$ in $A/\mathfrak{a}$ is a unit element of the algebra $A/\mathfrak{a}$. Therefore, the ideal $\mathfrak{a}$ is regular if and only if the algebra $A/\mathfrak{a}$ is unital. Suppose that these conditions are satisfied.

The mapping $\mathfrak{b}\mapsto \mathfrak{b}/\mathfrak{a}$ is a bijection from the set of ideals of the algebra A containing $\mathfrak{a}$ to the set of ideals of the algebra $A/\mathfrak{a}$. These are the ideals of the ring $A/\mathfrak{a}$ because this algebra is unital. Finally, by Theorem 1 of I, §9, No. 1, p. 115, the ring $A/\mathfrak{a}$ is a field if and only if it is nonzero and its only ideals are 0 and itself. The proposition follows.