Goldie’s torsion class 𝒢 is a
class of left R-modules closed under taking submodules, factor modules, extensions,
arbitrary direct sums, and injective envelopes. The corresponding Goldie torsionfree
class ℱ is precisely the class of left R-modules possessing zero singular submodule. It
is shown that 𝒢 is closed under taking direct products if and only if nonzero left
ideals in ℱ have nonzero socles. Another theorem gives four conditions equivalent to
the following: Any direct sum of torsionfree injective modules is injective. One of
these four conditions is that the ring R is an essential extension of a finite direct sum
𝒢(R) ⊕ L1⊕ L2⊕⋯⊕ Ln, where each Li is a uniform left ideal of R. It is
natural to ask when R actually equals this direct sum. A sufficient condition
for this to happen is given. Rings in which every torsionfree principal left
ideal is projective are studied. Particular attention is paid to those rings
whose Goldie torsion filters possess a cofinal subset of finitely generated left
ideals.
