Let R be a ring with Jacobson radical J (R). Given a left R-module M, a supplement submodule of M is a submodule K M for which there exists K ≤ M such M = K + K and K is minimal with respect to this property. In general, supplement submodules are strict generalizations of direct summands. Those rings for which every supplement submodule of a finitely generated projective module is a direct summand have been widely treated in the literature (see [6][7][8]11,13,16]). However, it is an open problem to give an internal characterization of these rings (see [9]). The purpose of this note is to give more results about supplement submodules in projective modules and determine their relationship with a generalization of projective modules (radical-projective modules). These results will allow us to generalize some characterizations of rings in which every supplement of a finitely generated projective module is a direct summand, to those in which every supplement of a (non-necessarily finitely generated) projective module is a direct summand.Throughout this paper, R will denote an associative ring with 1, module will mean left R-module and morphisms will operate on the right. J (R) will denote the Jacobson radical of R and R-Mod will be the category whose objects are left R-modules. Recall that a submodule K of a module M is said to be superfluous (written K M) if for everyIt is trivial that a submodule K of a module M is a supplement of K M if and only if K + K = M and K ∩ K K. For every module M, J (M) will denote its Jacobson radical and given x ∈ R, (x : 0) will be the left ideal of R {r ∈ R: rx = 0}.