Torsion-free modules of finite balanced-projective dimension over valuation domains

Let R be a valuation domain. Several characterizations are obtained of torsion-free R-modules having finite balanced-projective dimension (bpd) by using the new concept of an n-balanced submodule and a special pull-back diagram. Noteworthy among the various results are: For torsion-free R-modules M, bpd(M)≤pd(M), where pd(M) denotes the projective dimension of M. If M is a torsion-free R-module which is not balanced-projective, then bpd(M)=pd(M/B) for any basic submodule of M.