An useful tool in the study of the endomorphism ring S of a quasi-injective left R-module U is the duality defined by RUS which has been recently investigated in several papers ([7],[15]). The closed submodules of RU and SS (i. e., the annihilators in U and S of subsets of S and U, respectively) have a good behaviour in relation with this duality for they are precisely the reflexive submodules of U and the right ideals Z of S such that S/Z is reflexive. In [9 I it is proved that each closed submodule of a free right S-module sn has a supplement in sn and, as a consequence, that each finitely presented right S-module and each finitely generated submodule of a free right S-module have a projective cover. The first of these properties defines F-semiperfect rings, while the second one is called condition PH. As it is pointed out in [g], condition PH is satisfied by the following classes of rings: 1) semiperfect rings, 2) F-semiperfect right coherent rings and 3) right semihereditary rings. The endomorphism ring of a quasi-in-