The usual torsion theory for modules over an integral domain has the following well-known property: (A) Any direct sum of torsionfree injective modules is injective. This property does not always hold for torsion theories over a more general ring R.he main theorem of this paper determines nine necessary and sufficient conditions for (A) to hold in the setting of more general torsion theories. Inthe case that every module is considered to be torsionfree, then the conditions in the main theorem reduceto well-known conditions for the ring R to be left Noetherian. If a hereditary torsion theory satisfies (A), then its associated torsion filter possessesa cofinal subset of finitely generated left ideals. As applicationsof the main theorem, the torsionfree covers of Enochs are generalized to more general notions of torsion over more general rings, and then it is shown thatthe class of R-modules for which the torsionfree quotient with respect tothe Goldie torsion theory is injective forms a torsion theory if and only if property (A) holds for the Goldie torsion theory of R-modules. © 1969 by Pacific Journal of Mathematics.
