  Let A be a Noetherian local ring with canonical module K. We characterize A
when K is a torsionless, reflexive, or q-torsionfree module. If A is a
Cohen-Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K is
q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions,
we provide a generalization of Foxby's result to arbitrary Noetherian local
rings admitting the canonical module. In particular, since the reflexivity of
the canonical module is closely related to the ring being Gorenstein in low
codimension, we also explore quasi-normal rings, introduced by W. V.
Vasconcelos. We provide several examples as well.
