Corollary 1

— Let A be an algebra over a field K. For A to be a regular K-algebra it is necessary and sufficient for the ring $\bar{K} \otimes_K A$ to be an integral domain.

The stated condition is clearly necessary. Conversely, assume that $\bar{K} \otimes_K A$ is an integral domain and denote by E the field of fractions of A. By Prop. 4 (V, p. 141) the ring $\bar{K} \otimes_K E$ is an integral domain, hence E is a regular extension of K, by Prop. 9; by V, p. 141, Cor., we conclude that A is regular as K-algebra.