E-inversive semigroups with a completely simple kernel

We show that an E-inversive semigroup S has a completely simple kernel KS if and only if it contains a primitive idempotent (in that case, KS is the set-theoretic union of the groups eSe, where e is a primitive idempotent of S). Along the way, some equivalent conditions for a semigroup to be E-inversive are given. Moreover, some applications of the above theorem will be pointed out.