In this note we examine the
𝒯 -torsion submodule of quasi-injective R-modules, R a ring with unit, where 𝒯 is a
torsion class in the sense of S. E. Dickson. We show that for a stable torsion class 𝒯 ,
the 𝒯 -torsion submodule of any quasi-injective module is a direct summand, while if
𝒯 contains all Goldie-torsion modules, then every epimorphic image of a
quasi-injective module has its 𝒯 -torsion submodule as a direct summand. In addition,
we show that for a stable torsion class 𝒯−, all 𝒯 -torsion-free modules are
injective if and only if R = T(R) ⊕ K (ring direct sum), with K Artinian
semisimple.
