Definition 1

Let $k$ be a field and $A$ a $k$-algebra. One says that $A$ is absolutely regular (resp. absolutely normal)$^1$ if the ring $A_{(k')}$ is regular (resp. normal) for every radicial extension $k'$ of finite degree of $k$.

Every absolutely regular (resp. absolutely normal) $k$-algebra is regular (resp. normal), as is seen by taking $k' = k$ in definition 1. Recall that the $k$-algebras $A$ such that the ring $A_{(k')}$ is reduced for every radicial extension $k'$ of finite degree of $k$ are the separable $k$-algebras (A, V, p. 117, th. 2).