Idempotent separating congruence on a regular semigroup with a regular idempotent

Let S be a regular semigroup. A congruence ρ on S is called idempotentseparating if the associated projection homomorphism ρ # : S → S| ρ , isidempotent separating. Hall shows that if u is an idempotent of a regularsemigroup S then every idempotent-separating congruence on uSu extends to aunique idempotent separating congruence on SuS. An idempotent u of a regularsemigroup S is called regular if fuR fL uf for each f ∈ E(S). In this paper, weproved that if u is a regular idempotent of S then S = SuS. Also we find therelationship between the idempotent separating congruence on S and uSu, when uis a regular idempotent of S.