Let S be a right reductive semigroup. Then the semigroup S is embedded in the semigroup Λ(S) of all left translations of S as its left ideal. Thus we regard S as a left ideal of Λ(S). Then Λ(S) is an essential extension of S as a right S-system. By Berthiaume [2] there exists the injective hull I(S) of S containing Λ(S) as a right S-subsystem. In §1, we give necessary and sufficient conditions that Λ(S) equals I(S). It turns out that both left zero semigroups and right reductive primitive regular semigroups satisfy any one of these conditions. Consequenly we show that full transformation semigroups (written on the left) and the direct product of columnmonomial matrix semigroups over groups are right self-injective. We also study right nonsingular semigroups, semilattices of groups S which satisfy the condition that Λ(S) = I(S). In §2, we state some results on right self-injective semigroups. In particular it is shown that any direct product of right self-injective semigroups with O amd 1 is right self-injective. Consequently we show that any direct product of self-injective semigroups is self-injective.