Let (𝒯 ,ℱ) denote a hereditary
torsion theory for the category of modules over a ring R. In this paper the splitting of
projective modules is studied, and it is shown that this is not equivalent to the
splitting of quasi-projective modules. In addition, situations arising from the class of
torsion modules 𝒯 (or the class of torsionfree modules ℱ) being contained in the
injective or in the projective modules are considered, and several conditions sufficient
for an especially strong form of splitting are given. Finally when 𝒯 is closed under
injective envelopes the following is shown: every module splits if R is an artinian
generalized uniserial ring, and projective modules split if R is a QF − 2
ring.
