Example 2

Let $A$ be an artinian $k$-algebra. If $A$ is normal, it is reduced, hence isomorphic to a finite product of extensions of $k$ (A, VIII, § 8, No. 1, prop. 2). The following conditions are equivalent:

A is separable; — A is absolutely regular; — A is absolutely normal.

In fact, if $A$ is absolutely normal, it is separable; if $A$ is separable, for every finite extension $k'$ of $k$, the ring $A_{(k')}$ is reduced and artinian, hence isomorphic to a product of fields, and consequently regular, so that $A$ is absolutely regular.

If moreover the $k$-algebra $A$ is of finite degree, the preceding conditions are also equivalent to saying that $A$ is étale (A, V, p. 34, th. 4).