Remark

Suppose the $k$-algebra $A$ absolutely regular. Then the total ring of fractions $F$ of $A$ is identified with the product of the $A_q$, where $q$ runs through the set of minimal prime ideals of $A$ (IV, § 2, No. 5, Prop. 10); it is therefore a separable $k$-algebra. Conversely, suppose that $F$ is a separable $k$-algebra; for every minimal prime ideal $q$ of $A$, the $k$-algebra $A_q$ is a ring of fractions of $F$ (IV, § 1, No. 1, Cor. 3 of Prop. 2 and No. 3, Cor. 1 of Prop. 7), hence a separable $k$-algebra (No. 4, Prop. 6). If moreover the $A$-module $\Omega_k(A)$ is projective, it follows from Cor. 2 that the $k$-algebra $A$ is absolutely regular.