Rickart modules relative to singular submodule and dual Goldie torsion theory

Let [Formula: see text] be an arbitrary ring with identity and [Formula: see text] a right [Formula: see text]-module with the ring [Formula: see text] End[Formula: see text] of endomorphisms of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text], where [Formula: see text] is a fully invariant submodule of [Formula: see text], is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule [Formula: see text] of [Formula: see text] as [Formula: see text] and [Formula: see text], and investigate some properties of [Formula: see text]-inverse split modules and [Formula: see text]-inverse split modules [Formula: see text]. Results are applied to characterize rings [Formula: see text] for which every free (projective) right [Formula: see text]-module [Formula: see text] is [Formula: see text]-inverse split for the preradicals such as [Formula: see text] and [Formula: see text].