Several characterizations are given of (Zelmanowitz) regular modules among the torsionless modules, the locally projective modules, the nonsingular modules, and modules where certain submodules are pure. Along the way, a version of the unimodular row lemma for torsionless modules is given, and it is shown that a regular ring is left self-injective if and only if every nonsingular left module is regular.

Let R be a left and right perfect right self-injective ring. It is shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius. It is also shown that the same conclusion can be drawn if r(A ∩ B) = r(A) + r(B) for all left ideals A and B of R.

A left FPF ring is a ring R such that every finitely generated faithful left R -module generates the category of left R-modules. It is shown that such rings split into R = A⊕B, where A is a two sided ideal, and A contains the left singular ideal of R as an essential submodule. If R is FPF on both sides B is two sided too, and R is the product of A and B. An example shows this is the best possible and that right FPF does not imply left FPF.