  Let $T$ be a Noetherian ring and $f$ a nonzerodivisor on $T$. We study
concrete necessary and sufficient conditions for a module over $R=T/(f)$ to be
weakly liftable to $T$, in the sense of Auslander, Ding and Solberg. We focus
on cyclic modules and get various positive and negative results on the lifting
and weak lifting problems. For a module over $T$ we define the loci for certain
properties: liftable, weakly liftable, having finite projective dimension and
study their relationships.
