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. The main theorem of this paper determines nine necessary and sufficient
conditions for (A) to hold in the setting of more general torsion theories. In the case
that every module is considered to be torsionfree, then the conditions in the main
theorem reduce to well-known conditions for the ring R to be left Noetherian. If a
hereditary torsion theory satisfies (A), then its associated torsion filter possesses a
cofinal subset of finitely generated left ideals. As applications of 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 that the class of R-modules for which
the torsionfree quotient with respect to the Goldie torsion theory is injective forms a
torsion theory if and only if property (A) holds for the Goldie torsion theory of
R-modules.
