If \(T\rightarrow R\) is a ring morphism of commutative unital rings, then an \(R\)-module \(M\) is called liftable to \(T\) if there exists a \(T\)-module \(M'\) such that \(M= M'\otimes _T R\) and \(\text{Tor}^T_i(M',R)=0\) for all \(i> 0\); \(M\) is called weakly liftable to \(T\) if it is a summand of a liftable module. The author investigates concrete necessary and sufficient conditions for a module over \(R=T/(f)\) to be weakly liftable over \(T\), where \(T\) is a commutative noetherian ring with identity and \(f\) is a nonzero divisor on \(T\). Then, the author focus on cyclic modules and obtain various positive and negative results on the lifting and weak lifting problems. Finally, for a module over a commutative local noetherian ring \(T\), one defines the loci for certain properties: liftable, weakly liftable, and having finite projective dimension, and one studies their relationships.