QUASI-INJECTIVE MODULES AND STABLE TORSION CLASSES

In this note we examine the J-torsion submodule of quasi-injective R-modules, R a ring with unit, where J is a torsion class in the sense of S. E. Dickson. We show that for a stable torsionclass J, the J-torsion submodule of any quasi-injective module is a direct summand, while if J contains all Goldie-torsion modules, then every epimorphic image of a quasi-injective module has its J -torsion submodule as a direct summand. In addition, we show that for a stable torsion class J, all.J -torsion-free modules are injective if and only if R = T(R) ⊕ K (ring direct sum), with KArtinian semisimple. © 1969 by Pacific Journal of Mathematics.