For a subsemigroup T of a semigroups,we let Reg(T) and reg(T) denote respectively the set of all regular elements of T and the set of all elements of T which are regular in S.We characterize semigroups with Reg(T)=reg(T),where T runs over one of the following families of subsemigroups:left ideal,right ideal,and the insection of the right ideal and left ideal.