Drazin (2) has recently introduced the concept of a pseudo-invertible element of an associative ring or semigroup. In this note we first show that such an element of a semigroup S may be characterized by the fact that some power of it lies in a subgroup of S. The main result (Theorem 2) provides a new criterion for complete simplicity: a simple semigroup is completely simple if and only if all its elements are pseudo-invertible. This is applied to the principal factors of a general semigroup and two previously known results are obtained as corollaries.