A module as a torsion-free cover

LetR be a ring, and let (ℐ, ℱ) be an hereditary torsion theory of leftR-modules. An epimorphism ψ:M→X is called a torsion-free cover ofX if (1)M∈ ℱ, (2) every homomorphism from a torsion-free module intoX can be factored throughM, and (3) ker ψ contains no nonzero ℐ -closed submodules ofM. Conditions onM andN are studied to determine when the natural mapsM→M/N andQ(M)→Q(M)/N are torsion-free covers, whenQ(M) is the localization ofM with respect to (ℐ, ℱ). IfM→M/N is a torsion-free cover andM is projective, thenN⊆radM. Consequently, the concepts of projective cover and torsion-free cover coincide in some interesting cases.