This paper explores the structural aspects of epi-S-X-retractabe and S-X-compressible modules for a ring A with unity, a multiplicatively closed subset S ⊆ A, and an A-module X. It generalizes classical epi-retractable and compressible modules by establishing equivalent conditions under epimorphisms and direct product. The study derives criteria under which a torsion-free divisible S-X-cyclic module over an integral domain is simple, and provides necessary and sufficient conditions for S-X-compressibility, including monomorphism characterizations. The results extend to essentially and slightly S-X-compressible, injective, and quasi-injective modules.
Singh et al. (Fri,) studied this question.